Newton introduced the following ideas. A fluent is a quantity that changes. A fluctuation is the change in the quantity. If the fluent is designated x, the fluctuation is designated dot x. One problem is that the fluent is not well-defined because the amount of change can change. To address this issue, Newton considered the ratio of fluctuations. If x and y both changed, what is the ratio of dot x to dot y? The result is known as the method of fluctuations and was developed by Newton in the 1670s. For example, let x increase from x to x plus o find the ratio of change of x squared to x. So x squared will change from x squared to x plus o squared. So the actual change will be the difference to o x plus o squared. Since x changed from x to x plus o, the change in x is o itself. And so the ratio of change is to o x plus o squared over o or to x plus o. And so this is the ratio of the change in x squared to the change in x. And Newton goes one step further and he probably shouldn't have taken this step because it opens up a lot of difficulties later on. But it's only by pushing our boundaries that we learn what we can do. So the ratio of change in x squared to the change in x was 2x plus o. And we might consider the problem this way. This ratio of change depends on o, the amount by which x changed. And the larger the value of o, the greater the ratio of change. And we might consider this ratio of change as depending on two things. One is x itself and the other is o, the amount by which x has changed. And so Newton considered the last ratio, the ratio obtained when o is zero. You might view this as the change that doesn't depend on o. And if we let o equal zero, this is 2x. Now Newton gives a slightly different argument in the Principia, book 2, Lemma 2, where he considers the following problem. Let a rectangle besides a and b have a increase at rate a and b increase at rate b. Then the rate of increase of the area is ab plus ba. Newton's proof is the following.  And so there's our increase of a. And from b minus a half b to b plus a half b.  And so our change is going to be the difference. So we'll subtract the one from the other to get ab plus ba, which is what Newton claimed it was. Now if you understand the use of infinitesimals as we've presented them, you should be very suspicious about the use of infinitesimals as we've presented them. In 1734, the use of infinitesimal quantities came under attack by someone named George Berkley, if you're American, or probably more correctly George Berkley, if you're British. Berkley, by the way, has the distinction of attempting to form a college in Newport Road Island in the 1720s. That didn't work out so well in Berkley and his wife returned to Britain in the 1730s. In 1734, Berkley pointed out that there were several problems with Newton's approach using infinitesimals. First, in problems like this, x to the n and x plus o to the n can only have different values if o is not equal to zero. Okay, so that's not a problem because this allows us to divide, to obtain a ratio between the fluctuations, but then to find the last ratios, Newton set o equal to zero. And that's possible on the right-hand side, but on the left-hand side, we're dividing by o and o can't be zero. And so the first problem is that o is both zero and not zero. How about the argument from the Principia? Berkley pointed out that the argument in the Principia is clearly tailored to make the rate of change come out correctly. Newton knows he wants the answer AB plus BA and so he specifically chose this increase from A minus a half A to A plus a half A and from B minus a half B to B plus a half B. But this is wrong. The true way to find the increase is to allow the sides to increase from A to A plus A and from B to B plus B. And if we do that, the area increases from AB to AB plus AB plus AB plus AB. So the change in area is actually AB plus AB plus AB. So to get Newton's result, we have to ignore AB but we can't ignore AB no matter how small it is. As it turns out, Berkley was right. Berkley acknowledged that these methods obtained the right answer but the fact the methods were based on falsehoods and clever tricks suggested that this was mere luck or perhaps the compensation of errors. Ultimately, the problem is that the methods weren't trustworthy. You couldn't rely on them to always produce the right answers. And in point of fact, Newton probably knew that the functions had a very shaky basis which is why he usually avoided them in his mathematical works. In fact, the first accounting of these infinitesimal methods appeared in the Principia and later was elaborated upon in the optics. And in fact, Newton approached the problems of calculus in an entirely different fashion.


Okay, so our next speaker, Yan-Chi Chiu, I apologize for the pronunciation. Thank you. He will speak to us on Patterson's Elephant Construction for Point Processes and Reconstruction of Harmonic Functions. So first of all, I would like to thank the invitation from the organizers and it's a great pleasure for me to have an opportunity to speak our recent work with Professor Alexander Buffett-Darth. So we will basically, roughly speaking, we want to generalize classical constructing which is called Patterson's Elephant Construction in dynamic system to random setting, namely to the setting of a point process and use it to reconstruct certain harmonic functions on unidisk or even higher hyperbolic spaces, high dimensioning. So the motivation of this work has already been mentioned in the talk of Professor Alexander Buffett-Darth. So let us recall briefly these motivations. So we will consider the space of all a homomorphic function on unidisk. So then for those functions, which is not only a homomorphic but also with certain integrity condition, we obtain the classical Bergman space. And in particular, we are interested in the Hilbert space case for the index p equals to two. So then we call a subset of the unidisk uniqueness set for this Bergman space. If it satisfies that all function from this Bergman space is uniquely determined by its restriction to this fixed subset. For instance, if this subset has an accumulation point inside the unidisk, then of course by the classical result for homomorphic function, of course such a set will be a uniqueness set for the Bergman space, even for homomorphic functions. But of course we are more interested in this situation when the set has no accumulation point inside disk, but of course it could have accumulation points on the boundary. So basically we are considered infinite subset. So this means exactly has the same meaning that any Bergman function in this space, if the restriction to our set is zero, then it has to be identically zero. So this notion is closely related to a notion called the Bergman zero set, meaning that such set is exactly the location of homomorphic function in our space. So simple remark set is a uniqueness set for fixed Bergman space, if and only if it is not a zero set for such Bergman space. Okay, now we're going to consider the random set given as the zero set for this classical Gaussian analytic series. So we just consider a formal series and we consider the coefficient to be very simple, just the standard complex Gaussian, mean zero and variance one. And it's an elementary fact that these formal series, it has radius convergence one. So it is not just formal, it's in fact define a true homomorphic function inside the unit disk almost surely because it's radius convergence equals to one. Then of course we can consider the zero set of this homomorphic function. I will denote it by this symbol. So by a famous result of Paris-Virac says a following, so which is already mentioned in previous talk by Bouffet d'Auf that the distribution of the zero set of this Gaussian analytic function is exactly the determinant point process induced by this classical kernel called the Bergman kernel, which is in fact the kernel of the orthogonal projection from the L2 space on the unit disk to the space of our Bergman space with index two. Then as a colliery of our solution to this, I will not mention this solution to this line space as a complete conjecture. I think probably Bouffet d'Auf will mention this tomorrow which says that in fact as a general result for any determinant point process, once it is induced by the reproducing kernel of just some abstract reproducing kernel help space in some L2 space, then almost surely this random set will be the uniqueness set for the corresponding reproducing kernel help space. So in particular, since by this result, this random, this zero set of the random kernel function is the determinant point process corresponding to the Bergman space. So almost surely this set is a uniqueness set for the Bergman space. So in fact, this is the study of the uniqueness that is quite subtle. Let us just mention briefly, we are really in a critical situation. So it's kind of when you first see this, it's kind of a counter-induction. So you have a holomorphic function and you concede the zero set of this holomorphic function and the result is this zero set of certain holomorphic function cannot be zero set of other holomorphic function. But because here our random holomorphic function is given by this thing, so it can be proved easily that this thing can never be, almost surely can never be the Bergman space A2. Actually with a little computation, this can be shown to be A2 minus epsilon B. So it's the integrability of this holomorphic function is just a little bit less than the critical situation. But then usually in the study of a holomorphic function, we have usually encountered a situation that although you have a very badly behavioral holomorphic function, you can probably divide by some factor, this factor never vanish. So this factor never vanish so it does not change the structure of the zero set, but it will probably improve the integrability. So this, which means that probably, so the question is, we ask whether we can like find a factor from this holomorphic function and to improve a little bit the integrability without just drawing the structure of the zero set. But this calorie tells us now, so we can really not improve any integrability without destroying the structure of the zero set. Okay, so then this comes to our further step. In fact, the proof of this theory is really in a general setting and we do not use at all the holomorphic structure in this concrete situation. So it's a very natural question to why not find a direct proof in this concrete situation. So this is the motivation of this work. So just to recall that almost surely, this set is a uniqueness set for our Bergman space. What does this mean? This means that for a generic realisation, so you, with probability one, we fix a subset, fix a realisation, then any Bergman function in A2 is uniquely determined by its restriction to our set. So natural question is how? So this is just theoretically, so it's determined by restriction, but how it is determined? Can we find an explicit formula so that we can reconstruct, recover simultaneously and moreover explicitly all Bergman function from just the data which is spent as the restriction of the function to our set? And once this question is just, it's very natural to ask a similar situation, why not just ask this question for this particular random subset? We can ask this question for more general random subset, and we can also ask for this question for not only the Bergman space, but for probably other even Banach space of a holomorphic or even harmonic functions on unit disk. Then the first step, so yes, I will really try to focus, try to emphasise that the most difficult part of the work is really this work, simultaneously. Simultaneously means that, so you have, because this is improbability setting, we have a fixed realisation and this realisation works for an uncountably many function usually, so. But before going to that step, we first started even, this first step is not clear, so for a fixed function and for a fixed point, so this is not really a restriction, so we can, because we are a reconstruct a holomorphic function, once you can reconstruct a holomorphic function on a countable many points, so you can reconstruct the function all, so but the really problem is, so this is a drawback, so for a fixed, but let us start at this problem first. So for reconstruct a fixed Bergman function, we obtain the following results, saying that, so first we record the Lopachevsky hyperbolic magic and we have, this will be used later, the hyperbolic real area on the unit disk. So we prove that if we fix a Bergman function and fix this point, then using, so forget for some time this expectation on the denominator, so we obtain here, this is part of some discreet version of mean value, so this is weighted average, so this weighted average it's when, so for a sequence of weights, when this weight goes to the critical point as goes to one, then it is very close to the value of this function, so let us look at the picture of this formula. So we have a unit disk and a lot of set points, so this is our fixed realization of the point process, which is given by the zero set of, sorry, zero set of this holomorph function. Now we know the value of a holomorphic Bergman function on these points and we want to obtain the value of this point, which is probably not in our original set. So what we do is the following. So we first want to obtain certain, so basically speaking we are going to do the following. So we draw the hyperbolic disk and average, average using some weighted average of values inside this hyperbolic disk. And then we take, we let the hyperbolic unit disk to go larger and larger, then this value, the average will be closer and closer to the value at this center, origin of the disk. So as a corollary, this we have, this is just a Borel-Cantelli argument from the previous results. So we have exactly this, the value of our Bergman function on this position will be the weighted average of this thing. So there's some subtlety here. So I take the average really from any line and take the average here, the summation here, they take larger and larger. The fact we use this is because in fact if we use just one summation to take the sum over all points, then this is usually divergent. So this is really necessary to obtain. So this is in fact, when you find this result, it's very easy to check, but before you find this result, it's really difficult for us to realize we should do this discrete version, mean certain principle value procedure. So this is explained, so what I explained is really this. So that by summation is needed because in general, this is not known. Then this is as I said, this is just the first step. We want to construct just one function. Now, of course, one function you can use in just one, it's okay, almost surely, so you can use accountable in many functions, again okay, just by intersection. But the problem, the difficulty is really our formula is not continuous with respect to the Bergman norm. So you cannot extend this explicit formula just by saying, ah, okay, and this holds for then subset, then it holds for all space. This is just not true because this formula holds for instance for polynomials. So, but you can never hope that this can work for any, because polynomials can be dancing like AP space, not just two, you can work for a two, a four, a three, but this is just not possible. So then we want to analyze what is really going on in our proof. So in fact, this homomorphicity, we are really using the harmonic assumption, which basically we are using the mean value, area mean value property, or in certain steps we are really using the spherical mean value property of the function we want to reconstruct. And the function, the complex value function can really be replaced by vector value, or some Hilbert space vector value function. And so these two steps are just trivial. So you look at the proof, it's very simple. But these steps really non-trivial, less trivial generalizations, we can not only reconstruct the function for the Bergman space, but in fact we can reconstruct more. More in the sense that we have, so if you look at this space, it seems to be very close to this space, just a little bit less integrable. So because without this epsilon, so this thing goes to, when z goes to the boundary, this goes to zero. So with this weight, this, it will be more possible that this integral be finite. So defining this, of course, you have bigger space, and we add some this growing condition on the norm of our function with respect to this weight. And this small, this really very small gap, we achieve a little bit, will help us a lot later. Okay, now we come to the original question. Whether we can do simultaneous reconstruction for this thing, for this Bergman space. As I said, as a corollary of our solution to line space conjecture, theoretically this is true, you can reconstruct, but the fact here I said it's impossible to reconstruct simultaneous all function using actually linear procedure. So probably there exists some non-linear way of the construction all, but this result tells us, so not only using the exponential weight, so I use compact supported because there's some issue of convergence. So for simplicity, we work with compact supported weight, then any weight you use here, just for if you want just reconstruct one point, it's not possible by using this average method. So here this constant is really universal for all radial weight on the unit disk. So this, so when we see this probably we will even ask whether the supremum here it's measurable or not. So the measurability of this part or in fact the proof of this inequality is reduced to this factor. This supremum actually is the norm of the following function, the L2 norm of certain holomotiv function. So okay, yes. So now we have this result. We cannot simultaneously reconstruct our all-Bergman function. So it's natural to ask probably because we can reconstruct really the fixed function, probably we can reconstruct some smaller space using that method. So the smaller space, in fact, we obtain some abstract condition on the space so that we can reconstruct. So this space is just to give the several examples, which kind of space we can obtain the simultaneous reconstruction. So we work first with the weighted Bergman space with some rapidly growth in condition, growing weights, so think of this. So by this I mean some holomotiv function on the unidix, which is L2 integrable with a weight. So if this weight is the constant weight, this goes back to the original Bergman space. If this weight grows rapidly on the boundary. So of course this will force your function to be more regular because the integrability is better. You have these integrabilities with respect to bigger weight. So your original function should be, the growth on the boundary should be more controllable. But in fact, with this function, I will, why we really want to focus on Bergman space? In fact, there's one thing that I want to explain now is the following. So in fact there's this space, Hardy space, which is also possible to reconstruct our last, a substituation of the last case. So this is just those holomotiv function on the unidisk with A and ZN, with some square integrable coefficient. Then for such a very classical function space in functional analysis, we know that for any sector, from, for any sector, this is the rate, this is okay, this is one over square root of two. So for any sector of this, for almost all sector, you take the, there will be this limit. So the value on this, the value of our function on this position, then you take, you can take a limit here. Then you obtain a function on the boundary. And with this function on the boundary, you can solve digitally problem, then you will find the original function F, provided that this measure mu. So here I am really speaking of the, the big measure here, provided this measure is absolutely continuous with respect to the big measure. But however, even this procedure, if we consider singular measure, then although it can be, we can make a sense so that this has a meaning, that there's a boundary, boundary value, but this boundary value will be almost surely zero with respect to the big measure. So in fact, this procedure can never reconstruct the original function. So our result is the following. So if we consider the first two, the weighted Bergman space, or a general, more generally, some reproducing kernel helper space with certain assumption on, gross assumption on the reproducing kernel, then for all function in this space B, this partisan salivar reconstruction formula holds almost surely. So almost surely this holds for all function. And for the last space, which is a generalization of the classical hardy space, and the result can be improved a lot because here, this formula holds not only for all function, but also for all points. And also, this along a subsequence can be dropped. So it's just really a limit. So in fact, all this work, when I speak of the zero set of this random holomorphic function, it can be generalized to more general point process on the unit disk, such that the following assumption is certified. So first we needed some conformal invariance in average, meaning that the particle, the number of particles inside any disk, inside any subset is proportional to the hyperbolic area of this subset. And we need some small variance assumption of the linear statistic. So this linear statistics, the variance of linear statistics is controlled by the linear statistics of the square value. So in fact, all these assumptions are in particular satisfied by some point process and determine the point process with permission kernel, which we just mentioned, and more generally for negatively correlated point process. So in fact, this result is a little bit complicated. So in fact, we can also generalize this result in other direction replacing these... So if we really want to focus on reconstructed functions from this space, we need some, we need more dense assumption on the point process. So suppose our point process are more dense. So this assumption, if beta equals to one goes back to the original problem. If beta greater than two, then this is much larger than the zero set of this random holomorphic function. Then we do obtain a concrete formula for obtaining the original function. So there are some very natural problems arise in this work. In fact, it seems to us that even the existence of such, of the existence of subset of unidisk satisfying this formula seems to be new in the literature. But our works really prove the existence. So we really, it's really natural to ask whether we can construct a deterministic subset satisfying certain these kind of reconstruction properties. And it will be more challenging to give sufficient condition or some geometric criterion or a subset so that it will satisfy a given reconstruction property. So in fact, all this work, not it's not restricted to the unidisk, it can be worked on real hyperbolic space and complex hyperbolic, quaternion hyperbolic space and on these also locally finite, infinite connected graphs, et cetera. And in fact, on the level of major theoretical thing, so what we are doing here is formula if we understand these as some weight B, X, Z, that X. So if we look at not this function, we look at very small function, just the harmonic function, which can be extended to the closed unidisk. Then this formula says exactly that this measure almost surely converted to the Poisson measure, with respect to the point, harmonic measure with respect to that point. So using such, such result can be generalized to some more general situation for some negatively correlated, negatively curved remaining manifolds. And I will just mention that in complex hyperbolic space, the harmonic function will be replaced by the harmonic function with respect to the curved Laplacian. So this is the Laplacian for complex hyperbolic and this is the Laplacian for real hyperbolic space. So that is, yes, that is all. Thank you.


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So grab a snack, grab a friend, and hang in. Seven, six, five, four, three, two, one, go. Guys, welcome to today's vlog. Today, what we're going to be going over is a medium length shag haircut. Thing I like most about this is that we're going to do some layering techniques throughout the haircut to create those short layers. I think what people struggle with when they're cutting this haircut is that the layers get really heavy within it. Then we're going to blow it dry and what we're going to do is some dry hair cutting techniques that I know you guys are going to like. A lot of slide cutting in this to create that shattered edge. You can see this is the end result. We have a lot of layering in there. You can see the layering in the back and then this broken shattered baseline. And that's what makes this haircut so unique. These techniques are very salon friendly. You can use them every day in the salon. So let's not waste any time. Let's get started. Here we go with our step by step. So I want to start sectioning this haircut symmetrically. So what I'm going to do is a center parting all the way down center back. Then what I'm going to do is create a triangle section right in the bang or fringe area. And I'm going to go about two inches up from the forehead on that to the point of that triangle. So what this is going to do is give me a symmetrical section on the front of the head that I can create any type of bang that I want later on. Now we're going to create a horseshoe section off of the corner of that triangle. So it's a little bit higher than the corner. The reason I did that is I wanted to stay right along that parietal ridge area. Depending on the density of the hair is really how big you're going to make this horseshoe section. So if they have a higher density of hair, you're probably going to do a higher up U shape or horseshoe section because you want to really look at the top of the haircut as the roof.  So that's the only sectioning for the haircut. So it makes it pretty simple. So what we're going to do is start on the side of the head. I chose the left-hand side of the head. I'm going to take vertical sections or vertical partings throughout, about an inch wide. So because the density of the hair, like we talked about earlier, it's not super thick, so I can take inch sections. If she has thicker hair and it's hard to see your guide through it, then definitely break that up. But we're just going to take those inch sections, bring everything forward to a stationary guide, and cut it. The key thing here is the elevation that I'm working with. So think about, look at my fingers. Look how high I'm elevating the hair. It's pretty much parallel with the face. If I went any higher, it would remove more weight. And if I go any lower, it's going to start to build up weight. So it really depends on, do you want to expand the shape around the face or do you want to collapse the shape around the face? Again, that's based on density. So if they have a higher density, real thick hair, then you're going to elevate higher. If they have a lower density of hair, thinner hair, then you're going to keep your elevation low and build up a little more weight. And maybe get a little lighter towards the top, because even though you have a low density of hair, you don't have a lot of hair, doesn't mean you don't want layering and movement. It's just how much do you want is going to really change the look of the haircut. So you can see that face frame that builds up. That's what the shag part is all about. But what I didn't want is to only overdirect everything forward and build up a ton of weight in the back. The shag kind of has layering all over it. So what I'm doing is now I'm going to create some concave layers in the back. So I brought everything forward, pushed all the weight to the back, and now I'm going into the back and removing all of that extra weight that I created. So like I've talked about in a lot of past videos, if you haven't subscribed to this channel, I've done over 600 videos of different haircuts. And the key thing here is when you're cutting hair, you're pushing weight to a certain place to either do something with it later or you're pushing it somewhere for a purpose. In this case, I pushed it to the back because I wanted to remove it from the front. But now I've got to deal with the weight that's in the back. So I'm creating the concave layering. And the great thing about the shag haircut is all of the movement and the layers that are in it. So I wanted to keep this medium length. The other thing I want to show you guys, this is coming all the way back to the center. So everything I'm working with is a stationary guide. It was a stationary guide in the front, now in the back with the concave layering, I'm pushing everything to the center, back of the head to create that. So what that's gonna do is have all of the weight kind of meet right behind the ear. The thing that makes that okay is that we're going so high with the layering, it's not gonna be super heavy. The other thing I want you guys to notice is that I haven't really messed with the length of the haircut yet. We're gonna do the length of the haircut when it's dry. I wanna see the weakness of the hairline or the baseline of the haircut. I wanna see all of that dry. There's no sense in cutting it wet. So now we're gonna work our way into the top and what I'm showing you there is that this is also gonna be a stationary guide. Everything's gonna come to the center. So what I'm gonna do is follow the head shape. So I'm creating a rounded shape to the haircut using 90 degree layers. The 90 degree layers are gonna give me a seamless feel. There's no weight being distributed really. It's seamlessly passing those layers over the head. When you see somebody that has a really heavy weight line in their haircut, it's because the angle wasn't quite what they wanted. So you're seeing that weight build up. So what I'm doing is going through following the round of the head with my parting and then over directing it to the center back using that as a stationary guide. So there's a couple things you're gonna notice me doing throughout this portion of the haircut. I will comb multiple times and change the elevation of the cut because I wanna keep it at 90 degrees. I think a lot of people would just pull this up real quick, cut it off and then just move on. And then in the dry cut, they'd have to fix the weight that they had build up. What I'm doing is I'm combing multiple times creating that seamless feel and following that 90 degree point across the head. Head shape plays a key role in how this haircut is going to turn out. So because the head shape is round, I have to shift my angle every single time I comb to make sure that I'm at the angle that I wanted and 90 degrees is the key to keeping that seamless throughout. All right, so we're just finishing off that section. Look at how much over direction is happening. So I'm taking everything from that side temple area, bringing it around the head and cutting it in the center back over my guideline. Whenever you're cutting hair, you wanna think about, all right, if I hold it at this point, what is it gonna do when I let it go? Anytime you over direct something in an extreme matter, so we're over directing this all the way from the temple to the very back of the head. So technically that would be pushing a lot of weight to the front. Now, when you match that up with the right elevation, that's when it changes everything. So if I did that same over direction but kept everything low, maybe low crown area, occipital bone area, I would have so much weight pushed to the front that the haircut to me wouldn't really look right and the client would have a hard time with it. Because we're at that 90 degree point at the very back of the head, it is still keeping, by the time I get around to the side temple area and I over direct it over, we still have a high elevation, so it's still gonna keep the haircut nice and light. So you gotta think about a few things when you're cutting hair. Think about where you're over directing the hair to, what it's gonna do then, and also the elevation that you're working with within that portion of the haircut. So now, fun little trick with the bangs. What I did was I combed every, that whole triangle over, twisted it towards my body over top. That gives me that over direction and then I just cut it with a razor, which gives me a nice textured fun feel to the fringe. It's nice and light and also it's not a perfect line. So it gives me a little more freedom. I think that matches up really well with a shag haircut because of all the layering and texture in it. To cut that fringe, we use the carving comb, which is available on freesaloneducation.com. It's probably the coolest $35 tool you will buy because it's a comb attached to two different types of razors. You have one razor that cuts 100% of the hair, another razor that cuts 50% of the hair, and then you have a comb. You have a wide tooth comb or a fine tooth comb depending on which carving comb you get. So it's just a very versatile tool. You can use it while you cut hair with scissors. You can have all those tools in your hand at once or you can use it by itself. So it's pretty awesome. This is also the brocato vibra straight iron. The thing I love about this flat iron and what separates it from all the other ones I've used is it vibrates as it irons over the hair. So it creates less damage, less friction, and also a nice smooth pass through of the iron. It's digital, works really well. I keep it on 410 degrees. Usually when I'm working on my guests, if they have fine, fragile hair, I'll keep it more around the 390 area, maybe 370. It's a great iron if you're looking for a new one for your hair. Now, I wanna talk about the switch in scissors that I just made. So the very beginning part of the haircut, I did everything with my DB 20 scissor, which is my precision cutting scissor. It's a 5.7 inch scissor. Now I'm moving to my Mizetani puffin, which is a dry cutting scissor.  And I'm doing a slide cutting technique, which is a half close of the scissor, sliding the scissor down. Now, why this technique works and why I love it so much is because I went through and did all that layering so it skinned out the bottom of the haircut. So I really was able to see where I wanted to cut it dry and then just go through and slide cut and define that portion. You couldn't do this. You couldn't just start off by flat ironing somebody's hair without a haircut and then go through and just slide cut the bottom and make it shattered. There's a purpose behind everything that we do. So we remove the weight from the top of the head using the concave layering, allow it to fall down and then go through, do our detail work dry. And it just makes the whole haircut make sense. So I went through all the way around the haircut doing that slide cutting technique, just a half open, half close of the scissor, like I said. And it just pulls in that all of that texture, makes the layers have a little bit of separation and air.  You could see all of that texture in there, the face frame, how it works, it's medium length. It's such a great cut, hope you like it. Let me know in the comments below, thanks. All right guys, like always if you like this haircut then hit the like button, hit the share button, share this video with all of your friends out there. Also, if you have any questions, post those in the comments below. Also, if you're not a hairdresser and you're looking for a new hairstylist, check out our new website, hair salon locator.com. It's full of all of our FSE partner salons and stylists. So the people that watch these videos and train with us on a monthly basis are on there so that you can find them and find the right salon for you. So check out hair salon locator.com. Thank you guys so much for watching this video. I'll see you on the next one. Thanks. And guys, remember if you haven't subscribed to the channel, make sure you hit the subscribe button below because you could win this Vibra straight iron. Good luck. Let me know in the comments below if you've subscribed. Thanks. Hey guys, it's Matt back from freesaloneducation.com. I wanna invite all of you guys to join our new membership program. It's the FSE partner program on freesaloneducation.com. It's $19.99 a month. The great thing about this program is that you're gonna get an online class every month either by me or someone on the FSE team. Also, you get access to our super private FSE salon group on Facebook. On that Facebook group, you can connect with all the other members that are part of the FSE partner salon program. Also, you can ask me any questions that you want daily. And you're also gonna get added to our FSE partner salon and stylist locator so that people that watch our videos, we get millions of views a month, tons of people asking if we know of great salon. So that's really where this whole concept came from. So we're gonna help you train. We're gonna work with you in the Facebook group and we're gonna add you to the salon locator so people can find great salons out there. I look forward to connecting with all of you guys as you sign up. Thanks. So grab a snack, grab a friend and hang in. So grab a snack, grab a friend and hang in.


Good morning, saints. This is Brother Mike Sunday morning podcast, 9 a.m. Mountain Time and 11 a.m. Eastern Time. Thank you for joining me this morning. This is the Deep Things of God with Brother Mike. And I've got an interesting Bible study for you today. I think you're gonna find it fascinating with real life experiences attached to it. And I wanted to remind you of our ministry, hardcorechristianity.com. We can go there and see all of our ministry services. We have two live Zoom deliverance services every week, Tuesday and Wednesday nights at 6 p.m. Mountain Time. Send me an email, Mike at hardcorechristianity.com and I'll send you the ID and the password for those two services. They are quite remarkable. Lots of healings, lots of deliverances. Then we have two live services every week, Thursday and Friday nights at the Arizona Deliverance Center. We're downtown on 15th Avenue just south of Osborne Road. Those are at 7 p.m. Arizona and Mountain Time.  One of them is our main channel, which is youtube.com slash House of Healing AZ. You can catch those services live and the healing and deliverance portion of the services are also included on our streaming channel. So you can watch the deliverances actually occurring. You can watch the healings actually occurring in real time and you're good to go. Also on the website, on the homepage, you can catch all my local radio shows. I've been on the radio here in Maricopa County, Arizona for 20 years and I'm on KXXT Christian Radio 1010. The main broadcast is every morning at 7.30, Monday through Friday. I'm also on seven days a week and the schedule is on the front of the homepage on the website. All right, let's get to our Bible study. Got some interesting information for you today. Take a look at Luke chapter 13. Pretty shocking section of text here. Quite remarkable. It says here, Jesus told him a parable and he told him a parable of the fig tree. You remember that? The short version of that parable is this fig tree is not producing fruit. So the owner of the vineyard said, tear the thing down, get rid of it. And then the manager of the vineyard said, I'll tell you what, let's take a shot at salvaging this fig tree. We'll dig it up, fertilize it for a year or two and then after that, if it doesn't produce any figs, what we'll do, we'll cut it down and get rid of it, all right? And then the Holy Spirit then illustrates that parable with a real life situation. Right after that parable, it says Jesus was teaching in one of the synagogues on the Sabbath day.  The main Sabbath was on Saturday, but they had midweek Sabbaths, special Sabbaths, celebratory Sabbaths and so on. We don't know which Sabbath this was, but it kind of looks like it was the main Sabbath on Saturday. Remember Jesus was crucified on the cross of Calvary on the Sabbath, but that was a midweek Sabbath. He was crucified on a Wednesday, but traditionally they picked Friday and then you've got good Friday and all that stuff. That stuff was just all made up by religious people, it never actually happened. But it says here, behold, there was a woman in the synagogue that had a spirit of infirmity, okay? Now the Greek word for infirmity there is estonia. It's a generalized term for being weak or feeble and it could encompass any kind of, you know, condition. It's not conditioned specific. It's a generalized term, okay? But in this particular case, it looks like the disability was severe because she had had it for 18 years. 19 years previous to this service in the synagogue and her meeting Jesus, she did not have the condition. But what happened was 19 years earlier, a spirit entered the woman's body and it was a spirit of infirmity. Now these demons are stubborn as they can be and they cause a wide range of illnesses, sicknesses, diseases, disabilities, all of it. And once this spirit gets into the person's body, it doesn't have the power to instantly make them disabled or instantly kill them or instantly give them cancer to where it's so bad they die. It's a progressive system. It's a process where the demon gradually tears down the body. Now, if you see somebody with physical joint issues, almost 100% of those are spirits of infirmity. Rheumatoid arthritis, scoliosis, kyphosis, different, you know, joint disabilities. All those in my experience over the years have always been spirits of infirmity. Once you get the spirit out of there, the person recovers. And it says here, she had this condition for 18 years and she was bent over. The Greek word there is sunkupto. It means to stoop forward. So she was kind of a humpback, so to speak. And she could not stand up anymore, it says. Verse 12 says, Jesus took a look at her and he says to her, woman, you are loosed. Apoluo is the Greek word. It means to untie or unchain or release from being in bondage. You are loosed from your infirmity. Now, notice this. When a spirit enters a person's body, could be an infirmity spirit, could be a lust demon, could be a spirit of addiction. Once that spirit enters your body, it has the condition that you now have. So you don't need the name of the demon or the name of the sickness to get that out of a person's body. You don't have to have that. You don't need to have the names of demons. You can get them out, no problemo. But this woman's condition was orthopedic. And he says to her, you are loosed from your infirmity. Same Greek word, Esthania. Instead of actually naming the illness, I don't even know if they had a name for that illness back then, I'm assuming they did. We would call it rheumatoid arthritis or kyphosis or something. And he laid his hands on her and immediately she was made straight. Now that's a Greek word, an ortho, where we get our English word orthotics or orthopedic. An orthopedic surgeon, technically, is someone who straightens out your bones, so to speak. And that's what this is talking about. She was made straight. She suddenly just stood right up, which is a medical impossibility with that condition. And then she started glorifying God. Here you see the hidden secret of divine healing. The purpose of the healings are to glorify God. That's why you get healed.  But the end result of your healing and God's love for you is that you are to glorify him. You are to let your praise go. Crank it out. Well, then something bad happened and that's gonna happen to you. If you decide, and I hope you do today, to develop within yourself a healing and deliverance ministry, which every born again Christian can have who wants it and who will make sacrifices to receive it, Mark chapter 16. It says, these signs shall follow them that believe they shall lay hands on the sick and they shall recover. Here you see a perfect example of what Jesus was training his disciples to do. The disciples were all standing around watching this event take place and Jesus was demonstrating what he later on and Mark 16 and Matthew 28, he was gonna tell them to do. So it was all OJT. It was three or four years of Jesus being here on the earth in his ministry. And he was putting his disciples, the 12 disciples and all the other disciples, there were probably hundreds of them that were following him. He had them in an OJT program on the job training. Well, if you do what God wants you to do and you start to minister in the spirit and you start to develop a deliverance or healing ministry, which I hope you will, you gotta remember something. The church is going to turn on you and you have to be ready for that. You gotta be ready for that. The church is gonna turn on you. These demons that were in this poor woman got into her 19 years earlier, but I have seen numerous cases over the years where the spirits have gotten into wounds and entered into a fetus. And the fetuses are then born with congenital disabilities, very common. It happens all the time. But these spirits can still be cast out whether they got in in the womb or whether they got in 19 years ago, whether it got in one year ago, doesn't matter. The Holy Ghost is well able to smash this thing, no question. So this pastor gets on him and he starts telling Jesus, the creator of the Sabbath what not to do and to do on the Sabbath, not realizing that Jesus was the creator of the Sabbath and the Lord of the Sabbath. So he starts, you know, the janitor starts to tell the chief executive officer of the company what to do. Here you go. You shouldn't be doing this. This should not be done on the Sabbath day. We have other days where you are to come and be loose from your infirmities, but we don't do that on the Sabbath day. So anyway, this guy would have made a great, a phony, fake TV preacher. Verse 16, Jesus said, wait a minute here, shouldn't this woman being a daughter of Abraham whom Satan has bound for 18 years? You can see here clearly Jesus is saying, demons are a representative of Satan. They are his essence. They come from him. They obey him. They do what they're told.  Man, she was really hurt and in an enormous amount of pain. What Jesus says to the pastor, again, remember if you decide to get in to a healing or deliverance ministry, hopefully your starts today, you got to remember that sinners are not going to oppose that. They're going to actually look into it and be interested in it. Sinners are not your problem. The biggest problem are the churches. The churches that have post-stated themselves, the pastors who've got yellow streaks down their back that are looking for security and money and accolades.  They've ruined the gospel and they have helped turn America into a cesspool of sin. That's what America is now. It's a cesspool of stinking, wrought gut sin. It's utterly amazing and it's a hundred percent the fault of the church. Well, Jesus dresses down this pastor and explains to him, hey, what do you think you're doing here? You're off the track, right? And then it says his adversaries were ashamed. Okay, that's interesting, isn't it? Not only was this pastor Jesus' adversary, but there were several people in the synagogue that day that were tracking him. They were his adversaries. So you had a mixed bag there. You had true followers. You had the people there who loved Jehovah and they wanted the good things of God. And then you always have the tears mixed in there. There it is. You'd have the good and the bad and in the pastor's case, the ugly, but they're all mixed up. And so Jesus is ministering to all the groups. He singles out the pastor first, then the adversaries, it says, were ashamed after he explained to them that this woman was also a daughter of Abraham, just like they were and that they're hypocrites. And shouldn't she be delivered since the great physician is here today? Shouldn't she be delivered from Satan's bondage? Now remember, not all diseases or illnesses or sicknesses or trauma is caused by demons, but the net result of it is demons either cause it or exacerbate it. So if you get into an accident and you get injured, that's not a demonic illness. That was a trauma event. It doesn't matter. The demons use trauma events to accelerate them and worsen them and exacerbate them. They make it worse. So even if the spirit doesn't cause a sickness, they are frequently and often behind the sickness, making it worse and keeping it from healing. They block people's healings and the spirit of infirmities, they are absolute masters at it. Blocking your healing, nothing works. Physical therapy, cortisone shots, surgery, nothing. All medical treatments failed. That's when you know, you can be pretty sure that there's a spirit of infirmity behind this mess. And then after this event happens, it says, verse 18, then Jesus said to them, okay, this happened at the same service after this lady was healed and after she went through her praise period and after Jesus straightened out his adversaries and this pastor, he stayed at the same service and kept teaching, right? He was teaching after that. And then he explains something fascinating. He says, you put a grain of mustard seed in the ground and it starts growing. And initially it goes fast. And then eventually, if it's not destroyed, it becomes some great, gigantic fig tree. And then he says, the fowls of the air, the fowls of the air lodge in the branches of the fig tree. Wow, he's talking about that pastor and the adversaries that were at that synagogue service and he spoke a parable illustrating them. So he goes from the disabled woman to Satan to his adversaries all in the same service, all in the same service. They're the fowls of the air that lodge in the branches. I always remember, I try my best to get my YouTube listeners to develop what I used to call terror cells where you open up a small group of people in your church and it only needs to start scripturally.  And then after that person is delivered or healed then they go back and tell different peoples and pretty soon you've got this whole new ministry cropping up inside your church and that will go on for a period of time until you get caught and then they'll throw you out of your church. That's a good sign, definitely a blessing because that means God wants you to start your own ministry and that exact same thing happened to me years ago. I was at the Dream Center, which is a mega church. Somebody got mega church in Scottsdale and this exact thing happened to me, exactly happened to me and I'm at the Arizona Deliverance Center right now as a result of it and as I said you can watch our services every Friday and Saturday, youtube.com slash house of healing AZ and last night, two nights ago, Friday night service, excuse me, two nights ago a woman brought her disabled daughter to the service she was in a wheelchair and while several of my ministry partners were praying for the lady in the wheelchair I took the mother up to the front of the altar and then I found out that she had a lot of soul wounds and a lot of fears because of her daughter's future and her medical condition. She was scared for her daughter. Well, the Holy Spirit just jumped right on her and all kinds of spirits of fear and dread came out of her right at that moment. And then I went over to pray for somebody else and as I was walking by, I overheard one of our ministry workers praying against a bad back. I didn't know she had a bad back. So I came back over and I said to her, you got bad back? She said, yeah, I had a, I can't remember what it was, a fusion and some kind of a, I don't remember how many levels it was. Maybe she didn't even tell me. It could have been a two or three, what have you. She had had a low back fusion and she had a little altered gait. Her hips were tilted this way. Her back kind of bent that way and she was in chronic pain. Okay, so I prayed for her the first time and she wasn't healed. So then I prayed for her the second time and she wasn't healed again, strike two. So I figured, hey, this has got to be a spirit of infirmity. This isn't a, this isn't a disability.   The reason for that was she was starting to get shaky. She had been walking, trying to walk out her faith and get healed and it wasn't working. Her back was not healed. So I was holding her and I was kind of holding her up and I was praying for her at the same time. And while I'm holding her, I'm thinking in my mind, spirit of infirmity, you know, you've got to come out of there. This is it, I'm holding on, you're out. And suddenly it was weird. I felt something going up the woman's back and it went under my arms, up higher to her shoulders and went out. And as soon as it went out, the woman went limp and so I caught her. I was holding her up and she started crying harder. She was crying earlier because of chronic pain and fear. This sounded different. And I says to her, I think you're healed or something like that. And she's crying and it looks like tears of joy. And I said, go ahead and take a few steps and start walking again. She started walking and I literally saw, literally saw her hips level out. Oh, just like that, leveled out.  It was amazing. Other people saw it and the woman started walking around Friday night in the front of the altar and everybody started cheering and her healing got a standing ovation. Everybody was amazed and the Teen Challenge guys were there Friday night and they were flabbergasted as well. But again, the point is that every time you have the moving of the spirit, you're gonna have adversaries in the same service or you're gonna have the pastor or the church board or the elders or what have you, they're gonna be, some of them are gonna be against what you're doing. Okay. And that happens in almost every service. We don't have a lot of it at the Deliverance Center because people know what we do when they come there.   You always take them as learning experiences and then you just keep moving forward. Even if you falter and stumble, I've done it myself, of course, you get back up quickly and then you move forward. You're always moving forward.   Trust me, I know all about it. I'm an old man now. Or an older person, I should say. And I can remember just like yesterday when I was in college. I can clear as a bell. I can remember what I was doing, where I was decades ago. Time goes by so fast, you can't even believe it. That's why Paul said, you have got to redeem the time. Redeem the time. Now these spirits of infirmity, they are stubborn things. And prior to praying for this woman, she had gone through deliverance on the other side of the Deliverance Center. And I got her to forgive her ex-husband who was kind of had abandoned their daughter who was in the wheelchair of there. And she had built up some resentment and pain and disappointment over her ex, the daughter's father. And when she repented of that, evil spirits started jumping right out of her. It just instantly happened. They just started coming out. And so she had already repented of bitterness and unforgiveness before I gave her the bear hug. So the groundwork, so to speak, had been done. I said that to say this. Most of these joint issues I've noticed over the years are related to unforgiveness, bitterness, and ought. The Greek word is, I taste. It means to have a yucky feeling about somebody that you don't like or hurt you in the past. That's ought. It's different from unforgiveness because unforgiveness is an attitude issue. It's a mental issue. It's a choice issue. You can by faith, in obedience to God, forgive someone for what they did to you. But if you still have ought against that person, it can block your healing. And most of the people that you're working with, most of them that have joint issues, arthritis, fibromyalgia, I've seen, I don't know, 50 cases of that instantly healed. Fibromyalgia, those are usually caused by fear, spirits, rejection, demons, unforgiveness, ought. I've seen them instantly healed. Just it's very common to see fibromyalgia healed. But these joint issues, generally speaking, in my experience, they are related somehow to people that have some kind of ought or, you know, bitterness or resentment or bad memories or they rehearse negative thoughts in their mind when they think of that person or when they see that person. This ought thing can block your healing and it won't come out. You see that it's like a cork and a bottle and you don't get that new wine coming out.  You have to forgive that person and you have got to get that ought out of there. And Friday night, we had already done that with this lady. She had ought against her ex. And exes are extremely dangerous.  It used to be positive and it flipped to abuse and it flipped over to abuse.  They injure your soul, they damaged you because you took an offense. And that allowed joint problems to either start or exacerbate and they got worse. It's very similar to what Jesus said, you've got a church of believers, but in that group, in that church, literally almost every church, there are tears in there, tears, yike. And that's what bitterness and unforgiveness and ought is. It's a tear. Growing in the church, a weed, it's a fowl that lodges in the branches and it just hides there. Sometimes it doesn't see anything. That's what birds do. Sometimes they're in a branch and they're stone quiet. You don't even see them.  That's the purpose of a tear. It's a purpose of a fowl. And that's the purpose of bitterness. The spirit of infirmity uses bitterness and ought and unforgiveness as his fuel, as his power. And it gradually weakened this woman over 18 years. Can you imagine that? 18 years, she's down here. She was there 19 years ago, but the demon had taken her through this process, 18 years of it. Now, obviously if you're in an abusive relationship right now, the council dozens of peoples over the years, if there's a safety issue involved or a severe emotional abuse case, you've got to get some separation. You've got to temporarily separate.   But either now or later, you are gonna have to face it. Whatever they did to you and it doesn't matter what they did to you. What's the worst thing you could have done to you?   Okay, either now or later, you got to get this through your thick noggin, as my granddad used to say. You're going to have to forgive that person and you're going to have to remove that ought from your soul. Your soul is where your emotions reside. And your soul is where the wounds and the hurts reside. The spirit man is where your relationship with God resides. The anointing, the Holy spirit, it's all in your spirit man, but your soul man is the one that damages people. And the devil loves to use your soul to damage you. He does it all the time, right? And if you have ought against someone, if you have unforgiveness against somebody, the best time to get it out is on the Zoom services on Tuesday and Wednesday night. You don't even have to show your face on the Zoom. While they're praying, while somebody else is getting healed, you just follow the same pattern on your own. You put your hands on your body and you repent. You confess your ought. You confess your unforgiveness. That's an absolute requirement with God. You have to confess it out loud. Confess it out loud. Then you gotta repent of it and ask for forgiveness for it and apologize to God for what you were carrying in your body that allowed that spirit of infirmity to enrage itself and bind your joints. In some cases, these demons are so powerful that they bend your joints like they did to that poor lady in Luke chapter 13. This is how the system works. This is how you can get healed. Friday night, after that lady had gotten healed, she was crying, tears of joy, and all she kept saying was, thank you, thank you, Jesus, thank you, Jesus. And what was the purpose of the healing? Because when the woman in Luke 13, when she got healed, she stood straight up, she glorified God. That's the purpose of it because, obviously, the great physician heals. I don't have no healing powers. I'm fresh out of them. In fact, I never had them. Let's say I have a little secret there. That's a secret. I never had any healing powers. I don't have nothing. The whole thing is Christ the healer. It is the great physician. And I did the same thing Friday night, this exact process I went through. I propped the lady up, so to speak, in the front of the church. I announced that she was now the Arizona Deliverance Center's worship leader, and she wants to take the rest of the people there through worship. And she went with it wholeheartedly and excitedly, praising Jesus and everybody else was praising. And I was praising him, obviously, and thanking him for this wonderful healing. Now, you have something wrong with your body, particularly a joint issue. Okay, it's probably a spirit of infirmity. What happened was you tried to get healed using every method you possibly could.  Okay, now it's a red flag that you tried medicine, physical therapy, surgeries, medications, what have you, nothing's working. If nothing is working, now not every illness, obviously, is a spirit of infirmity, everything's not demons. But if you've tried everything else and nothing's working and you see your joints kind of morphing, they're bending, clunk like that, there you go. That is a spirit of infirmity. That's a spirit of infirmity. You can get him out of there. I think he came out Friday night, it was really something special. How do you do it? You just do it like this. Lord Jesus, so and so abused me and degraded me and hurt me, and I took offense at it, I developed ought over it, I had bad feelings for that person, I don't like them, and I repent of it today. I repent of it and I ask you, I ask you to forgive them and to bless them.  And if you can do that sincerely or keep doing it until you are sincere about it, that spirit of infirmity had a grip on you, but it's loosening as you're praying. It'll loosen as you're praying. And the next step is, you use your authority that God gave you. Mark 16, Matthew 28, Luke 24. It's all right there. Matthew 18, Luke chapter 10, all these verses are screaming at you. You have the authority now to get that spirit of infirmity out of there. If you have repented, if you have not repented, you're gonna have it until you're lying in a care center somewhere and people are coming to say hi to you every once in a while, most of the time they don't even come back, but that's where you're gonna end up in a care center. And God's not God's will, not at all. Friday night was part one on divine healing and I gave numerous revelations, many of which no one's ever heard before. The next seminar is part two of divine healing. That is something you're not gonna miss for December. Tremendous insights into the mysterious world of being healed. And one of those insights I've shared with you today, it is the spirit of infirmity.  You can be crying tears of joy like this lady in Luke 13 or the lady Friday night at the Arizona Deliverance Center. It's the same Holy Ghost that performed both miracles. It's the same Jesus that did both miracles. The one he was actually present there in human form, obviously he's not like that anymore, but the Holy Ghost was there in Luke 13. He was there at the Deliverance Center. Either way, if you repent and you forgive and you get that ought removed out of there and you cast that ought out of there and you renounce it, you turn against it.  Or a parent or a current spouse actually. That happens quite often too, as you all know. Once that's gone, why not do an idea? I just grabbed the woman and the spirit in her body apparently somehow felt uncomfortable with me grabbing her. Something happened, the anointing was moving. She was maybe believing in her mind. Something happened, I don't know exactly what happened and that thing just came up, went north and out and she was healed.  Okay, that wasn't a muscle strain or sciatica. That was a fusion that got healed. It was a remarkable miracle. You can see it on our YouTube channel right now after the podcast if you wanna go watch it. You can hear me talking to her. You can see my mistakes. You can see the things I did right. I leave my mic on so everybody can have a learning experience. That's important too. If you go to the youtube.com slash house of healing AZ, you have got a shot at getting that spirit of infirmity out of your body, all right? Please come and see me at the Deliverance Center. We're downtown on 15th Avenue, just south of Osborn Road. Please contact me on my email, micathardcorechristianity.com. Remember the Bible requires you to confess your sins and repent of it. The Greek word is metano eil. It means to make an about face. If you're still doing the sin, you haven't repented. But brother Mike, I repent every day. No, you're being deceived by demons. You have not repented. You did not repent. Repentance means to stop doing it and go back to the other direction. It was a term used in the military, metano eil.  You got to repent of your sins. If you keep those negative emotions, if you keep saying those negative things, if you keep rehearsing over and over in your mind, those negative thoughts, you are bound. You will never be delivered. You will never be healed, okay? I'm warning you, your blood is not on my hands, okay? But if you repent and you change, you're getting healed. No question. See you next time.