So, if we are saying that A is diagonalizable, it means that it admits the basis right, so the first thing. The way we will go about this proof is not by showing this, this, then this, this and then this, this. We are going to go about it in a cyclic fashion. We will show 1 implies 2, 2 implies 3 and finally 3 implies 1 and we will be done with it. That is our line of reasoning. That is actually the minimal effort that you can put in right, because you have 3 conditions okay. So what is this? A is diagonalizable, A is diagonalizable. This implies there exists B, a basis for V such that, such that what? Okay, now I am going to use the hybrid notation okay. You might think A was a matrix, what is this nonsense about putting in a box bracket, but let us just say A is the operator. As I said I will not distinguish between it. So then A represented in terms of the basis, which then becomes a matrix. We do not confuse this okay. I know it is a bit confusing, but I just want to avoid writing phi every time. Ideally we have used phi for the operator and A for the matrix, but let us just say for now, I mean I am tired of those Greek letters, so let us get back to English okay. So this is what it is. And what is this? This is basically A V1's representation in terms of the basis, likewise till A Vn's representation in terms of the basis where the B is V1, V2 until Vn, is it not? Yeah. This looks like the following matrix, some lambda 1, lambda 1, I do not know some size of lambda 1's. Let us say this is d1 cross d1. From here on I will just write lambda 2 I d2 until lambda k I dk, where I is the identity of size dk cross dk here. So this representation turns out to be something like this. So this is a given. If A is diagonalizable, then of course there exists some such basis subject to which if I represent it, it is going to look something like this. I can order the basis suitably so that all the lambda 1 related terms suffers, then the lambda 2 related terms and so on. It is just an ordering. Because I have an existence of the basis, I can figure out which are the ones that are coming from lambda 1, which are the ones from lambda 2 and so on. If this is given, I have to show that this is true. Very straightforward verification. What is the characteristic polynomial of this now? It is nothing but determinant what? I am going to just ask for a very preliminary knowledge of determinant here. What is this? X i minus this matrix, is it not? So this is X minus lambda 1 to the power or rather X minus lambda 1 I d1, X minus lambda 2 I d2 until X minus lambda k I dk, agreed? Now what is the determinant of this? This first block, X minus lambda 1 to the power dk. This one is X minus lambda 2 to the power, sorry d1, this one is X minus lambda 2 to the power d2, so on. This is just product X minus lambda I to the power di i going from 1 through k, which means that the algebraic multiplicity of lambda i is di. What I have to essentially now show is that the geometric multiplicity must also be di and will be done until this point there is nothing non-trivial going on, I mean as in non-trivial calculation, right? This is straightforward. I am just asking you to get the determinant of a diagonal matrix, okay? So maybe I will just keep, I want to keep that statement here. So I will probably erase this first part here. Now think about wi. What is this wi? It is the kernel of lambda ii minus a.  I am not making any distinction between the operator and the matrix. I hope that is well understood by now, all right? Sometimes I might omit that box, you might understand from context it is a matrix. What is this going to be looking like? This object inside, this is going to be the kernel of, look at the matrix. This is just lambda 1 minus, sorry, lambda i minus lambda 1 id1 until the ith block turns out to be 0 times idi, right? And every other block is lambda i minus lambda k. Of course, needless to say that these lambda k's are distinct. Why would I otherwise write them as up to k? I would have written them as n, right? So these are distinct, all right? So let's just put it there, lambda i's are distinct. It's only there because they are distinct that I am able to call this as the algebraic multiplicity. What do you think is the dimension of the kernel of this? What is the rank of this matrix? It is n cross n. How many linearly independent rows or columns are there? It's a diagonal matrix after all. Apart from these di rows and columns, everything else is full rank. It has a non-zero at a very unique position. Every one of those rows and columns have non-zeros at unique positions. So definitely then the dimension of wi is equal to di is equal to geometric multiplicity of lambda i. Earlier we saw that this was the algebraic multiplicity. Now we see that this is the geometric multiplicity. So now we have to show that 2 implies 3, right? So we are starting with the condition that the algebraic multiplicity and the geometric multiplicities are going to be equal. So now suppose algebraic multiplicity of lambda i is equal to geometric multiplicity of lambda i for all i belonging to the set 1, 2 until k where there are k distinct eigenvalues, right? What does this immediately bring to our mind? From this summation algebraic multiplicity of lambda i is equal to summation di. What is this going to be equal to? N of course because it is an nth degree monic polynomial, right? It is an nth degree monic polynomial. That is what the characteristic polynomial is because this is nothing but the degree of the characteristic polynomial, right? It must be the degree of the chi. This must be equal to n. So summation di is equal to n which means what? This also means from this equality that the summation gm of lambda i is also equal to n which means that summation dimension wi is also equal to n. Now the claim is that these wi is all finite dimensional vector spaces of dimension di. So they will have a basis each of size di of cardinality di. So now we are going to use that earlier result that we have proved in the previous module. Consider bi as a basis for wi then by our previous result which I am not going to write again. Union bi is a basis for summation in fact a direct sum of wi that is what we have just proved the precise the result just before this that is exactly what we have painstakingly proved somewhat, right? So that is true. What is this after all? This is a subspace residing inside v of dimension n, v has a dimension n. So this must be equal to v, yeah? Does anybody have any doubts about this? Again please feel free to ask. What we are saying is this is given to us by our definition the algebraic multiplicities must sum up to n the degree of the characteristic polynomial and since the geometric and the algebraic multiplicity of each eigenvalue is the same therefore the geometric multiplicities of all those eigenvalues i is equal to 1 through k must also sum to n but that just means what that the dimensions of the individual w i's must also sum through to n. So therefore I am saying cook up basis for w i's individual w i's and you can always do that because these are all finite dimensional with dimension d i and take the union and just a while back we have shown that if you cook up individual basis for w i's and take their union you get a basis for the sum of those subspaces. In this case the sum of those subspaces by the dimension matching here turns out to be nothing but v because v is n dimensional any n dimensional subspace sitting inside another n dimensional subspace must be equal to the originals vector space itself. So what are each element what can you say about each element inside this set it is an eigenvector is it not I do not care for which eigenvalue but each object inside. So since each element in union b i is an eigenvector yeah and also we know that this is a linearly independent set by our previous proof so I am not going to use it I am not going to prove it again we already taken the trouble of proving this and union b i is linearly independent what do we have 2 implies right summation d i oh sorry yeah that should be k thank you yeah please ask if there is any doubt about this proof the fact that so we have shown that 1 implies 2 we have now shown that 2 implies 3 what is left to be shown is in fact 2 implies 3 is the trickiest part I would say not tricky but because we have done the groundwork it seems very brief if I had not done the groundwork that entire proof of that previous module would have come in as part of this proof you understand right so that is why I built it up I just sort of split it up into small little proofs so that in this case the proof does not grow very big alright now we have to show that 3 results in 1 but I would say that is very obvious is it not if you indeed have n linearly independent eigenvectors I do not care for which eigenvalue and whatever ordering and whatever you choose right so maybe I should erase this part yeah that is the oldest it seems alright so what we have is that a has linearly independent eigenvectors a has n linearly independent eigenvectors say v 1 v 2 through till v n since these are all eigenvectors therefore there exist lambda I not all distinct necessarily okay such that a vi is equal to lambda I vi now I am going to just stack it up together as we have done so often v 1 v 2 until v n so I am not now treating this like a matrix if you want to treat this like an operator again you have to take the individual representations and see the action of a on v 1 so if you call it and so maybe I will do that right after I am done writing in terms of matrices that is nothing but v 1 until v n times lambda 1 lambda 2 lambda n which essentially means that if I call this matrix v then it is v inverse a v is equal to lambda 1 lambda 2 lambda n okay if I want to be a little more sophisticated with the operator theoretic language I basically say that a v 1 it is operate its representation in terms of the basis then a v 2 its representation in terms of the basis until a v n its representation in terms of this basis what does it look like well this is now just giving it a coordinate representation so it is an ordered basis now right so a v 1 is what lambda times v 1 but lambda times v 1 represented in terms of an ordered basis where the first basis in that ordered basis is v 1 so it is just 1 0 0 0 0 so lambda 1 times 1 0 0 0 0 until n is it not in other words this is equal to lambda times just v 1 b then lambda 2 times v 2 b until lambda n times v n b what is v 1 in terms of the ordered basis where v 1 is the first element the matrix representation it is the it is just the principle no principle like you know basis vector 1 0 0 0 0 so on so this is nothing but okay let me draw a line here this is nothing but lambda 1 times 1 0 0 0 0 then this is lambda 2 times 0 1 0 like this until you have lambda n times 0 0 times 1 which is nothing but just this so whether you want to look at it as an operator or a matrix it matters not this clear right so we have shown now that 1 is equivalent to 2 2 is equivalent to 3 and 3 is equivalent to 1 and at this point we might have kind of closed the story on eigenvalues and eigenvectors and say that yeah I mean most cases you will encounter operators or matrices where even if the eigenvalues are not equal I mean even if the eigenvalues turn out to be equal maybe due to some perturbation or some numerical error here and there it is not a big deal in most cases you are likely to encounter matrices or operators where eigenvalues are only distinct and you will be right in inferring such I mean if you think of it like a probability problem where there is n squared numbers you have to fill out in an array you have to actually be very careful in choosing your numbers so that the eigenvalues actually repeat what are the odds that if you pick out a polynomial at random with some random coefficients that you will actually encounter a polynomial with repeated roots it is very unlikely right so you might say in most cases in my life I will probably never encounter this well that is not quite true firstly as I said in some physical systems modeling yeah it is not some numerical calculation that we have in mind always some physical system which has repeated modes whether you like it or not by the very design see for example that bridge which collapsed at its resonant frequency right it is a physical phenomena after all that number is just incidental but it has a resonant frequency it was badly damped so when soldiers were marching on it the bridge just collapsed because it maxed with the natural frequency so there are things there are physical phenomena which we have no control over so by the very nature of some circuit or some mechanical system it may so happen that the system is bound to have repeated eigenvalues and in such cases you cannot just say oh I will just tweak this number a little by some epsilon and make the eigenvalues distinct and then diagonalize it and say all that is the no as it turns out that such small changes in that matrix yeah and that x dot is equal to a is the dynamical system we had that small change might actually change the nature of the solution quite drastically right for those of you who have actually solved differential equations and have encountered this you would appreciate this that if you have repeated eigenvalues right and quite often you will see that the solution is not necessarily of the form e to the lambda 2 lambda t alone but rather some t times e to the lambda t right so just that eigenvalues being coincident do change the nature of the solution right so that is the reason why we will invest ourselves even though you will see that for much of the next few lectures we will just be trying to get our way around this Jordan canonical form just to motivate why we are putting in that effort yeah that is the motivation so what if these conditions are not met and you do end up with a system which cannot be diagonalized what is the best possible thing we can do we cannot just throw up our hands and say no no deal with it it is a 10 cross 10 system or a million cross million system so what we will then try to do is get it down to its bare minimum essentials maybe I cannot diagonalize what is the best thing about diagonalizations completely decoupled again in terms of differential equations you had nth order differential equation you reduced it to n first order differential equations maybe not first order but at least we know our way around second order differential equations so suppose we cannot diagonalize something but let's say there is some sort of a special basis which allows us nonetheless to block diagonalize a matrix so that the smallest block is still one but the largest block is maybe no greater than two that's also a deal right that's also winner that the block diagonals so let's say it's a 10 cross 10 matrix and maybe some of those blocks are not diagonals I mean some of those entries are not just diagonals but at least the biggest block that you have the block diagonal that you have is of size 2 cross 2 it means that the highest order of the differential equation you need to solve is just second order you can still decouple the system in some sense right if you are not getting the picture let me just draw it here so suppose you have a matrix that looks something like this all right everything else is 0 so normally this is what one 6 cross 6 so it's a sixth order differential equation probably being represented by this a matrix right if you have x dot is equal to this times x but on the other hand if you have brought it down to this form it turns out you can treat this part and you can partition this x into just 2 cross 1 here 1 1 1 and another 2 cross 1 here so you can treat this like 3 first order differential equations and 2 second order differential equations yeah okay yeah 2 sorry so here it is yeah okay getting my counting wrong now right so but that's still a winner right because instead of having to solve a sixth order differential equation you're going to solve at most the largest differential equation is the most complicated one perpetually is a second order differential equation and then this is completely decoupled from the rest so is this so if you can get it to this form that's also a nice thing right so in our bid towards a form that resembles this we will now study something which initially will look like completely disconnected something completely disconnected from what we have been doing now all right but we will see the connection very shortly so I'll just name that object in the previous sorry yeah no no if the lambdas are distinct all if the lambdas are all distinct that is there are n distinct eigenvectors then of course it's diagonalizable but in the previous proof we have said it's k k is less than or equal to n so you don't need a distinct eigenvalues that's what we approved as long as those kernels have sufficient size so if your eigenvalues are repeated a certain number of times but the corresponding kernels also have that dimension to match the number of times it is repeated in the characteristic polynomial then we are safe even if the eigenvalues are repeated we don't care about it yeah so we are now going to focus on something called an a invariant subspace so simply stated it is this suppose you have v sitting inside which is a subspace w of course a is finite dimensional vector space in general we'll take it to be n the size of the dimension of v all right for a which is an operator from v to itself suppose a v or maybe we call it w a w belongs to the subspace w whenever w belongs to the subspace w all right then w is said to be an a invariant subspace of v all right what it means is the action of a on w does not take you anywhere outside the subspace w so the action of sometimes in for notational purposes people say that a big w is contained inside w which means that you take fellows inside the subspace w let a act on them now look a is an operator from v to itself so there was a chance that if you take an arbitrary random vector inside w and a random operator an arbitrary operator then there is no reason to expect that after the operator's action has been performed it will still live inside w it can be anywhere in v which is not necessarily part of w but this is a very special kind of an a invariant subspace subject to a given operator yeah if you can find a subspace such that the action of a on any vector inside the subspace still contain still lives inside this same subspace okay now it's all very well and good to say that oh this seems like very difficult object to wrap our heads around do these objects even exist yeah so I will give you the simplest possible example of such an subspace and don't laugh at this but okay so this is a invariant is it not so at least it exists always another well let me cheat in the sense be very lazy of course v is its own subspace by definition it is a invariant okay so that doesn't make you happy I am sure you're asking for more better examples I at least guaranteed to you that it's not a meaningless pursuit but you might still feel it's meaningless because these are of course I mean obviously they're going to be a invariant yeah exactly every sub every vector space is its own subspace so the whole vector space is of course a invariant because the operator is defined on the vector space itself that's kind of obvious so these are very lazy examples so when we come back in the next module I'll give you some more interesting examples which should hopefully get you more excited about the these kind of subspaces a invariant subspace sorry projection map yeah it's image it's invariant to its own image yeah it's I it's I'd important that's what yeah yeah that's a non-trivial that's one non-trivial example yeah but we will we are not particularly now focusing on the projection map that was kind of done in duster we'll see more important objects because remember our study of a invariant subspace is now with a particular goal in mind the goal is if you cannot diagonalize it get it down to its best possible block diagonalization in the smallest possible blocks we are not happy with just a big block because that still entails the solution of a higher order differential equation right so you want to get it down to as small as possible and these are the objects that we expect to help us right so when we come back in the next module I will talk about a few more examples of this and that should get us started on this


That is, I picked up the important structures in the human brain, and I mentioned which are the RTA supplies of those. So let's quickly go over there. Medulla, branches in the vertebral artery, the BICA, and the pons for the basal artery, midbrain, PCA, cerebellum, all the CAs, BICA, BICAS, thalamus, anterior thalamocortis and anterior corority from internal carotid. Posterior thalamocortis, posterior carotid, from AC. Please note the word, halamu, for fraity. It should ring a bell by now. They produce an arteries. They can produce acrylurin parts. When you see the word, fraity, it should ring a bell. Corpostrate, antrolateral central arteries, from MCA, antromedial central arteries, from AC. Corpus callosum, antiaphobic, ACA, posterior cartid, by AC, in a PCA. Next, hippocampus, hippocampus artery from PCA, as well as coroidal branches from IC and PCA. Coratix is the lateral ventricle. I think I've mentioned this many times. Coratix is the third ventricle. I've mentioned many times. Coratix is the fourth ventricle. I've mentioned many times. And finally, internal capsule, supermuscle, anterior limb, ACA, genome and posterior limb, MCA. I mean, these are so much must-know that I cannot emphasize enough. Now, let's come to the next subtlety of this chapter. Circle of Willis, which everybody is aware of, but there are many small small points integrated within the circle of Willis. So let's take a few words about the circle of Willis. Location, it is located here. Where my index finger is pointed. It is located in the intraperular possa of the brain, at the base of the brain. It is in the sister, intraperinecular sister. Or it is also reported that, cruel sister. I hope everybody's here in their mind what the meaning of the word sisterness. Yes? It is in a large southern prime space. The phyla is a surface of the brain. And the CSF is the interspace of the phyla. So the circle of Willis basically is a communication between the anterior circulation and the posterior circulation. The anterior circulation in the 2D is the intercarotid system. And the posterior circulation is the vertebral basilar system. Which are the participants of the circle of Willis? Little bit of the basilar, the 2BCA's, the ACA, little bit of the intercarotid, and the ACOM and the PCOMs. They are the participants. Very important. MCA does not participate in the circle of Willis. MCA is nowhere in this. MCA goes out like this. It does not participate. Branches of the circular Wilis. I'm going to repeat all the same branches, which I have mentioned so many times. Antidimedial central branches. These are the perforating branches, which come from the ACA part. Pustimedial central branches coming from the PCA. What do these branches do? Why are you saying that this is really my heart? They penetrate the brain's substance to the respective perforated substances at right angles, and they are end arteries. They do not enastomose. The perforating arteries, they are antometrial central, and the pustimidal central, they are central branches. They penetrate the brain's substance at right angle to the anterior perforated substance and the posterior perforated substance. I write angles to the brain's substance surface, and they do not enastomose on the death. They are central branches. They are penetrating end arteries, And their occlusions produce lacunar strokes. I hope everybody is absolutely clear about these. These are the branches which we have mentioned under the respective parties. Now I'm talking about the point of view of circular limits. These same branches also form small aneurysms in high-potency patients. Those are called microaneurysms or mid-ary aneurysms. And we will see, give them a name tomorrow. They are called HUSHA aneurysms. And they can rupture these E plus cerebral aneurysms. So this is the significance of these perforating end arteries. What are they? They are intramedial and the posterior central artery. Continuing with the circular limits. So this is the anterior circulation. This is the posterior circulation. In a normal patient, normal person, not patient. In a normal person, there is no intermixing of blood from the anterior to posterior and from the right to the left and vice versa. Each remains in its own territory. The vertebro basilar remains in its own side. The vertebro basilar of the side remains in its own side. And here intergrotted on this side, intergrotted on this side, they remain in their own territories. So you can divide the whole two sprains into four parts. This is the right upper, left upper, left lower, A of the R remains there. Pressure of the anterior and the posterior become equal in the posterior communicating artery and the blood flow stops. It won't make sense. Pressure from the right and the left become equal in this a-con and the stop there. So you're supposed to ask me, when to the next? A remains only when there is an occlusion. Then one side takes over the other side. And the classical example of occlusion that we shall see tomorrow is subclinic steels in Rome, which I have not elaborate now. I'm going to reproduce the same picture tomorrow. And I'm going to go into it more detail. So for this just to show you by way of example, the circular wheel list is there only for our protection. Normally blood does not mix. Most people, 75% of the cases, they mix only when there is a requirement. When there's an occlusion of this side or that side, are activated from the sphere or whatever. And that is the reason why we said that if there's an occlusion of the ACM before the A-con, there will be no problems because blood will go from this side to this side. But if there's an occlusion of ACM, this side to the A-con, there will be a problem. Here, now everybody's understood why. Likewise, if there's an occlusion of the PCA, approximately to the PCOM, there will be no problem because blood will go from here to there. But on the other hand, if there's an occlusion of the PCA, this side to the PCOM, this is the importance of knowing that A1 versus A2, P1 versus P2. If everybody understands this, if there's any doubt, please tell me, please try and repeat this point. If there's an occlusion of P, P1 here. What will happen? Blood from here will flow and supply this. Yes or no? But if there's an occlusion of P2 here, there's nothing to supply blood here. Likewise, if there's an occlusion of A1 here, blood will flow from here and supply this to the part. But if there's an occlusion of A2 here, there'll be nothing to supply the blood there. Didn't you get this point here? So, occlusion of P1 and A1, there'll be no problems of this number, A2 and P2. So important relationships of the circular bilis. All summarized in this picture. I'm going to go into the details tomorrow. Look at the location of the circular bilis, it's been enlarged in this picture here. It is located in the independent system. It is related to the pituitary gland. It is related to the cavernous sinus. Because of the location of the intergrotted artery in relation to the optic hyacinth, we have seen so many still roads. It's a lateral nasal hemianopia. Look at the relationship of the anterior communicating artery to the upper part of the optic plasma. It will produce white and brown water and an ovarian. Next, let me tell you something which I'll be which I've mentioned to you in the beginning of this chapter. I have enlarged that picture here. Can you see the acorn here? And can you see a small branch in the region of the acorn coming out from ACM which actually runs platforms like this to give the name? What do we call it? We call it the recurrent artery of bubner. We also call it the distravelous triad artery. The importance of this small artery. This is a small branch which supplies the medial PFC and it supplies the little bit of the anterior part of the striator. Acorn is the most common sign of aneurysm. When we are trying to get an aneurysm, here it says you. As an iatrogenic problem, you may not be, has happened quite often. In fact, it is well documented. Because of the proximity of the recurrent artery of bubner, this can also get kept in order in the beginning. When we are putting a line of clip to the aneurysm, because it's arising very close, it can also be accidentally kept. And if you keep one side, you will produce an aneurysm. Our aneurysm will be kept. And if you keep both sides, you will produce an extreme case of aneurysm in room which is known as akinetic bubner. But the person will become immobile and will not be able to speak. That is called akinetic bubnerism. So these can occur as an iatrogenic mechanism when you are trying to clip an aneurysm. It's a small artery, but it occurs. That is the significance. Now let's come to the next chapter. A few words about the cerebral circulation of the aneurysm. This is where I'm going to answer a few questions. In the cerebral circulation, there is very limited anastomosis. Very limited. A little bit of anastomosis takes place on the surface of the body. So there's a posterior one there. Between, I'm not going to put the picture here, because we have to show the picture of it. Between the branches of the ACA and the branches of the PC. A little bit of anastomosis takes place. The second place of anastomosis, as we have already seen just now, is this, that we already said. But we all want to care for it. What about anastomosis on the cortex and anastomosis in the brain, in the depths of the brain? In the depths of the brain, there is no anastomosis. I think I've said it already 15 to 20 times. The infinity n branches are non-anastomosis. Therefore, the story is over. The penetrating n branches do not anastomose. They are n-articles. What about the cortical surface? The cortical branches. On the cortical branches, in the gray matter, there is a very limited amount of anastomosis on the cortex. A little bit of anastomosis does take place between the cortical branches and the outer three to four millimeters. This is the answer to your question. Tomorrow, we will see an entity called ischemic penumbra. I won't elaborate on that now. That ischemic penumbra is nothing but, after an attack of stroke, there will be a few cells on the periphery of the stroke which will be some lethal injury, which will not be totally dead. That is called ischemic penumbra. Ischemic penumbra is one of these. It will limit the amount of cortical circulation. That is the only circulation that's only little bit of anastomosis that takes this very small amount of corticals on the surface. And what is left? I cover what ischemic penumbra. Only left ischemic penumbra. It's in the various branches. It's in the various branches. It's in the various branches. Very little. Mostly between the branches of the same arteries. That's what brings us to the end of the water-shedding part around this medium for the top of the world. But just know that you live in a water and anastomosis takes this little bit of corticals. Circulation of the gray matter is much more than the circulation of the white matter. Because of the neurons which you're going to have in your metabolism.  You'll notice that there are areas with a greater metabolism up here, like red and yellow. And those areas with lower degree of metabolism, they are more violet or blue in color. So that red scan is the best indicator of the metabolism or the circulation. Partics has got a much higher circulation. That brings me to what are the factors which aid with the cerebral circulation and what are the factors which inhibit cerebral circulation? This is important. Factors which aid the cerebral circulation is the most important is the systemic blood pressure. And the factors which inhibit cerebral circulation is the cerebral vascular diameter or other constriction of the cerebral circulation and intracranial pressure. And of course, this causes cerebral blood. You'll know more about that. So let's take these three entities one by one. Systemic blood pressure, cerebral vascular diameter and intracranial pressure. Let's take them and put a little bit in the detail. Systemic blood pressure. This is the main purpose for which determines the circulation of the brain. That's the reason why you have something called syncope.  One was vasodegal attack. Our symmetric mediated brain credit cardia in the case of low sympathetic vasocastric blood. Then it was against the shocking news or something. Suddenly there is vasodalytation. Suddenly drop of blood pressure. Not that the pressure to supply blood to the brain was the collapse. That is one example of acute vasodegal attack. Another example of syncope is chronic orthostatic postural hypotension.  Their speed is fooling. Decrease venous enter to the heart. Decrease cardiac output. Drop in blood pressure. Less cerebral rotation. The person drops down. So therefore, the most important in this regard, cerebral circulation is, propulsively, cancer is provided by the systemic medial blood pressure. Within certain limits, there is a systemic or in the blood pressure, like for example, a normal person, a healthy young man or any person. When you're lying down, if you suddenly get up, you don't get postural hemorrhage. But it happens in old people. That's why old people are told that when you get up on the bed, don't immediately stand up. You lie down, get up, put legs down for some time, wait for a few seconds, a minute, and then slowly stand up and catch a little something to make sure that you give time for the systemic sympathetic tone to pick up. Because I have seen old people lying on the bed and slamming in the bed. I've seen people getting injured in the head and all. So what happens in a young person? Why doesn't the young person fall?  When the systemic blood pressure falls, there may be some short within limits. In most people, as a reflex mechanism, the systemic muscular, the cerebral muscular resistance also decreases. So then, with the low blood pressure also, the cerebral circulation is maintained. So this is an important compensatory mechanism which takes place in young people. Let's come to the role of cerebral vascular habit. It's very easy to understand. If the cerebral vasoconstriction is there, there will be less cerebral pollution and vice versa. So what are the determinants of cerebral vascular habit? Let's take the most unimportant one first. Sympathetic tone. In the case of cerebral circulation, sympathetic vasoconstriction, it does not play major. That's why I just mentioned it in the beginning. The most important determinant of the cerebral vascular diameter is the pH and the ionic concentration. Specifically, hydrogen ion concentration of the CSF, pH, and PCO2. Increased PCO2 and increased hydrogen ion concentration produces cerebral vasodilatation. Must know. And I'm going to give you clinical use of this point just now. Increased hydrogen ion concentration increased PCO2. Increased VO2 is a cerebral vasoconstriction. And this point is used in clinical situations. For example, when we get cases of patients presenting with head injuries or after several strokes or brain tumors with increased intracranial pressure, what do we do? We have to develop artificial ventilation, intermittent positive pressure ventilation. And by means of that artificial ventilation, it is called mechanical ventilation, it's called IPPV, intermittent positive pressure ventilation. We actually regulate their PCO2 level. We control their PCO2 level to keep it at this level. It is 30 to 35 millimeters of work in 4 to 4.70 of us. Because at this level of PCO2, the cerebral vessels are in the right diameter to provide cerebral therapies. So this is our aim when we put up a criminal artificial ventilation of the head injury or any means in case of PCO2. Because PCO2 is a very important determinant of cerebral circulation. PCO2 is an hydrogen ion concentration. And likewise, increased VO2 is a cerebral vasoconstriction. What is the role of central tumor receptors? They do not have any direct role for the circulation. They play a role in the stimulation of the respiratory system. Central tumor receptors which are located in the medulla mostly, there are many of them, the useless factors that give us a lot of interest. What do they do? They respond to C-S-F-D-H, PCO2, and RTL-PCO2, but not RTL-PCO2. Central tumor receptors do not have U2 receptors. They are the main driving force to respiration at C-level atmospheric pressure on the normal cell surfaces. But they do not have any direct bearing on circulation, but they control the respiration. That brings me to the role of intractable pressure. The third one. Yes. I just have a question. For what innovation and control that C-O2, is the main reason to control that that C-O2? If you keep the C-O2 at this level, several vascular diameter is right for coffee. If it is below this level, several vascular pressure will take place. And if it is more than this level, too much vascular dilation will take place and that also is not good. Yes. Even with the increased P-O2? Yes. P-C-O2 is the main determinant. I told you. P-C-O2 is a more important determinant than P-O2. So it is pH, it is hydrogen concentration and pH are opposite of course. It is H-I concentration and P-C-O2 are the most important determinant. What about the role of intracranial pressure? Intracranial pressure, it actually hinders circulation. But yesterday I told you the normal intracranial pressure is around 10 millimeters or more. That's one hundred and fifty millimeters or more. Obviously, the mean arterial pressure that is the difference between the average between the systolic and the dastroic is much about this to maintain circulation. Now let's take a situation where the intracranial pressure rises above the mean arterial pressure. That will again produce the next component. Increased systemic blood pressure will activate the carotid sinus and aortic arts reflex. Finally, because of compression on the brainstem, respiratory centers, respiratory rate will also become irregular or slow and so on by our cessation and you get pushing stride. So this is the full mechanism of pushing reflex works and how pushing can be done. This is the role of intracranial pressure on cerebral circulation. Methods of determining cerebral blood flow. When I talk of cerebral blood flow, I mean mindful circulation. FMRI and TECS scan either of the nine factors of cerebral blood flow. For example, if you ask the patient to focus on a particular object and look at something, you can detect the cerebral blood flow in the possible. You ask the patient to think of something and you find no blood flow in the cerebral blood flow. Not only that, and so on so forth. But those are, just give us a semi-quantitative method. Suppose you want to get an accurate quantitative estimation of the cerebral blood flow. How do we do it? We do it by means of radioisotropic studies. And the two isotopes which are used for this purpose are radioisotope xenon gas, which is inhate. Or radioisotope krypton, which is injected, interoperated. In another case, you record the decay of radioactivity from the brain by means of a bigger counter. And the rate of decay gives you an estimation of the blood flow of the brain. And by this method, this is the normal blood flow of the brain. This is milliliters for a gram of brain tissue for a minute. This is how the blood flow is happening. So we have finished with the essence of the arterial supply of the brain, the structures, the words about the cerebral circulation, determinants of cerebral physiology, and the methods of determining cerebral blood flow. Now, let's come to a few quick words about the linear system of the brain and then the spinal cord arteries. Many of these veins, we have already dealt with in our Men-G's chapter. So I'm going to go through the quickies, only the ones which we have not dealt and talk about them. The brains of the brain, then they divided into superficial and deep, or external and regional, working the same. So let's quickly take the superficial and external veins. The brain's drain in the frontal and the parental regions. They are called the superior cerebral veins. And we have seen all these veins, yes or no? These superior cerebral veins, there were eight or ten of them on either side. They all drain by means of the bridging veins. I have already told you in great detail what the bridging veins are. You should know that by now. They drain by means of these bridging veins into the superior's and outer sinus. There is one big vein which joins the superior with the superficial and external veins, which is another superior anastomopic vein. The veins from the inferior part of the cortex, the temporal region, they all drain through the tensile sinus. And again, there is one big anastomopic vein which is called the inferior anastomopic vein, okay. This vein that you see here on the surface of the lateral feature of sylvius, on the surface, please don't add, I'm repeating the word surface. That is called the superficial middle sectional vein because it's on the surface of the vein. Where does this one drain? This connects these veins and at the same time it also drains through the plasma sinus. Superior, surgenal sinus and the cavernous sinus are the two most important areas. The venous sinus is to change the entire supra-lateral and infra-lateral surface of the vein. The other sinus is that we studied that they all drain the deeper parts of the vein. Let's see what the deep surface is. The deep surface, the deep venous system. The deep venous system also refers to the internal. The deep surface, let's take them one by one. Keep the arteries in mind. You have seen anterior cerebral artery, you have seen anterior cerebral artery. So with that, this is the anterior surface of the brain. This side, the temporal lobe has been removed to see from the surface. This side, the whole temporal lobe is there, you can see it. Take a look at this artery here. You should be able to recognize this artery. This is the, this is the MCA, I don't know if it's the MCA, I told you it goes naturally through the lateral feature of sylvias, yes or no? And this is the ACA. Why are we supposed to be able to identify MCA and ACA? I want you to identify and be able to clearly locate the location of this. So accompanying the MCA, there's another vein here. That's the deep middle cerebral vein. Why deep middle? Because there is already a sub-official middle cerebral vein which I mentioned just two minutes ago. This is running in the depths of the lateral feature that's what it's called, the deep middle cerebral vein which accompanies the middle cerebral artery. Accompanied the ACA is a vein which is called an anterior cerebral vein. It does not require much of a brainer to understand it. So the deep middle cerebral vein, anterior cerebral vein and there's a vein which is not shown here. These veins are all personalized. These three veins you like to form this major vein which runs around the cerebral blood vessels. Roughly parallel to the PC. Remember PC also runs around the cerebral blood vessels? So these three veins which is the third one is not shown here. They unite to form the basal vein. Basal vein. Why is it called the basal vein? Because it runs around the basis of the actual life of the tracery canal. These two basal veins, on the other side, they unite behind the mid-brain to form brain, cerebral, brain, vein. They unite. They all drain into the, they don't unite. They drain into the brain's cerebral vein, they drain into the brain's cerebral vein. So this is one set of deep veins. Deep in the cerebral vein and just a little bit. Now let's take a look on the cerebral vein. This is a little complicated, but we'll make it simple. So let me get this picture here, and let me take this one here. Take a look at this picture and follow the text there. Because this is the best way I can make it simple. This is, again, a surface of the brain with the superficial structures removed completely to show you only the govite lexes of the natural ventricles, the roof of the third ventricle, the temporium cerebellum. First get the orientation. Take a look at these veins, which are coming from the spriot veins. Yes, these are known as the thalamose spriot veins. And there are veins which are drained in the govite lexes. They are called the govite. So this is draining the thalamose and the spriotome, and this is draining the govite lexes. Both of them unite in the region of the interventricle of one row to form the internal cerebral veins. So same thing here, thalamose spriot veins, so the thalamose and the govite spriotome, and the coradil veins, unite in the region of the interventricle of one row, to form the internal cerebral veins. So one internal cerebral vein of this side of the middle, it wants an intercerebral vein of this side of the middle. Both these run from NG here to posterior. NG here to posterior on the roof of the third ventricle and I did mention that they run on the first ventricle They run like this and at the region of the pulvinar of the thalamus, they unite to form the brain, cell, brain, body. So they unite to form the brain, cell, brain, and body. And this is the vein which receives the two basal veins. The brain, cell, brain, and body then unites with the inferior societal science. It is there in sinus, which all of us have studied. And this union occurs in the palcotentorial junction. And this union occurs in the palcotentorial junction at this level. And this is the palcotentorial contluence, the straight sinus. And the straight sinus, then we have to give that the left transverse sinus. We have all seen this in our viewer channel. So this is how the deep cerebral vein and the bridge cerebral vein are formed and how we create the cerebral vein. That's why I told you in the beginning of this section that the superior sedentary sinuses and the cavernous sinuses drain the superficial parts of the brain and other sinuses drain the deeper parts of the brain. The deeper parts of the brain are formed by these veins. These take a good look at the internal cerebral veins, how it forms the great cerebral vein of the brain, and also take a good look at how these veins drain in the brain. And finally, do not forget how the great cerebral vein of the brain unites with the ISS on the straight sinus. It's like that to use as the left transverse sinus. So this is the simplest way to understand the neural cerebral vein. The rest, this is good to look at the picture, do not memorize what is written there. This is the base of the posterior cerebral vein, all very good to these veins. Now let's come to the last part of this chapter and we are done with the circulation of the spinal cord. We did touch upon it in the beginning of this class. When I was talking about the vertebral artery, I told you, the vertebral artery gives us two, the first two branches, the PCA and the posterior spinal artery and the anterior spinal artery. Let's go a little deeper. The PSA and ASA, the two PSAs and the one ASA. Contrary to what we may think, they are not very big arteries. Please get this point very clear. And they are not sufficient to supply blood to the entire length of the spinal cord. Therefore, sometimes the spinal cord is classified under two raw heads. One is the longitudinal riboarticles, which we have just now named. There's two PSAs, one ASA. I'm going to go into that before I do. First, the second set of branches, arteries with the spinal cord are known as the segmentary arteries. What do they mean by segmentary arteries? Segmentary arteries means that every segment of the spinal cord, a series of arteries will enter from both the cells. These aid an abet and enhance the circulation of the posterior arteries. We have to have a spinal cord as the two sets of spinal cord. A series of longitudinal arteries and a series of segmentary arteries must get this point straight. Now, the segmentary arteries that we have seen here, they are again further subdivided into two subgroups. A set of these segmentary arteries are called just radical arteries. What do radical arteries mean? They supply some of the spinal root regions. That's why radical arteries are used. And there are another set of these segmentary arteries which are bigger. And they supply a little more of the spinal cord. They are referred to as segmental medial arteries. So with this background, let's go deeper and take them on our way. The posterior spinal arteries. You've already seen it comes out from the vertebrate. Each side remains separate. It runs in relation to the dorsal root of the spinal cord and runs down. This posterior spinal artery. It supplies only the posterior one for the spinal cord. This posterior spinal artery is not a very major artery. And it is especially deficient from the segments T1 to T3. Therefore, in these segments, it requires reinforcement from the arteries which I told you just earlier. The segmental medullary artery. From this, it follows that if you do any reason, there is an injury to the posterior spinal artery in the region from T1 to T4, especially if the segmental arteries in this region are not strong enough. The patient is more likely to have ischemia, the posterior spinal artery syndrome in T1 to T3. This is the region of supply. T1 to T4 segments of TSS syndrome are more common. Now let's take the anterior spinal artery. The anterior spinal artery again arises from the vertebral arteries. And both the sides unite. And they run down in the anterior-medium fissure of the spinal cord. They do not run down to the anterior-medium fissure. Embedded in the biamid. All these arteries, they run down. As it runs down the length of the spinal cord, it gives us branches through the anterior-medium fissure which supplies and these branches are known as the sulcal arteries. To be sulcal arteries, it supplies the anterior two-thirds of the spinal cord. What does it mean? It supplies both anterior commissure, anterior white matter, most of the lateral white matter, anterior brain harm. It sends the periacmedial, okay.  It just does not supply the posterior uniculus problems. The anterior spinal artery is again deficient in T4 segment and L1 segment. And it requires reinforcement of the segment arteries. And if the segment arteries in this region are not sufficiently strong, then we can get ASS in row, which is more common in T4 and L1 segments. Finally, anterior spinal artery also gives branches to the medial part of the medulla. And we have already seen that these branches are known as the paramedian medullary arteries. So if these anterior spinal arteries are occluded in the medulla part of the medulla, they can also produce medial medullary arteries. I have not written it here. Now let's take the next segmentary arteries. I told you the segmentary arteries can be subdivided into two groups. The vaticular, the smaller ones, and the segmentary medullary, the larger ones. Let's take a look at the same picture. These vaticular arteries, they arise from big named arteries in each segment of the human body. So in the cervical region, they arise from the ascending cervical and deep cervical branches of the vertebral artery. And they go to each segment of the vertebral artery. In the thoracic region, they arise from the posterior intercostal arteries, which are branches of the thoracic vertebral artery. And they enter into the spinal cord to each vertebral cortex. In the lumbar region, they arise from the lumbar arteries, which arise from the lumbar artery. And they enter into the sacral region, they arise from the lateral sacral artery, which is the branch of the internal artery. So at every segment of the spinal cord, they arise from different named arteries. The ascending deep cervical, the posterior intercostal, the lumbar arteries. At every segment, they enter through the intervertebral forearm, and as you can see here, they enter through the intervertebral forearm and they divide into a posterior root, an anterior division, and they supply the region of the roots of the spinal cord. Important point to be carried over. These radicular arteries are not very big arteries. They supply only the region of the spinal nerve roots and a little bit of the spinal cord here. And these radicular arteries, they do not, I repeat, they do not, anastomose with the spinal artery, anterior and posterior. You can see here that's an ascending. This is an important point to be noted about. Radicular arteries do not anastomose with the anterior and posterior spinal arteries. They just add to the blood supply, but they do not anastomose. Because I'm going to say something different just after this. Coming to this next group of segmentary arteries, the larger ones. And we gave them a name. We called them segmentary medullary arteries. How do those segmentary medullary arteries go? They also arise in the same way. They also rise in the same way. They also enter in the same way. They enter through the intervertebral program. They also divide the same way, posterior and anterior division. They also divide the same way. You can see in both these pictures. But there are some important differences. First, they are much bigger compared to the radicular. Can you see? They are much bigger. But they supply a larger part of the spinal cord. And these take a very good hard look at this picture. At this picture. They anastomose with the anterior and posterior spinal arteries. So not only do they supply a larger part of the spinal cord, they also anastomose with the anterior and posterior spinal arteries. So they enhance the supply of the anterior and posterior spinal arteries. That is why I told you a little while back that the anterior and the posterior spinal arteries are not sufficient by themselves. They need reinforcement for the segmental medullary arteries. Segmental medullary arteries anastomose with the anterior and posterior spinal arteries and they help to enhance the circulation. That is why I told you again, let me repeat, if the segmental medullary arteries are deficient in the T1 to T4 segment, then you can get a posterior spinal operation syndrome and if there are deficient in T4 and L1 segment, you can get. Finally, one more point to make. We are just almost there. There are just two more slides that we have done. These segmental medullary arteries and the radicular arteries, they are both not present in the same segmental. If one segment has got a radicular artery, it will not have a segmental medullary artery. And if this particular segment has a segmental medullary, it will not have a radicular artery. So both segmental medullary and radicular arteries are not present together in the same segment. Either one or the other will be present, but not both. Therefore, they are irregularly and often at the same time. They are not both present at the same time. Another point, there is one particular segmental medullary artery which is a very big one. It is a grand end one. And let's see. There is a separate name for the great anterior segmental medullary artery. Adam Key, this. This is a very big segmental medullary artery which is usually present on the left side. In the lower paracolama region.  That in most people, it supplies the spinal cord between the cervical and the lumbosy. It has got so much of blood supply. And in some people, it can be the main source of blood supply between the lower two parts of the spinal cord. It becomes even more important than the spinal artery. It is a great segmental medullary artery. So these are the facts about the spinal cord surgeon. ASA, PSA, radicular arteries, segmental medullary arteries. And the great segmental medullary artery. Finally, the spinal cord has been removed. Yes. Is that the one I can use? I am going to mention. These are the ones which we have. Last one, spinal cord weights. Nothing special to mention about them, except a few salient points. All the venous veins in the spinal cord, they drain into three sets of veins, anteriorly, and three sets of veins, posteriorly. Three veins anteriorly, three veins posteriorly, vice versa. Three anteriorly, three posteriorly. So there are six longitudinal spinal veins. All of them, longitudinal spinal veins, they drain into the nexus of veins which I have been saying repeatedly here, having my head. They all drain into the internal, vertebral venous plexus. I have reproduced this picture here and this picture here, all of you are clear in your mind where is this internal vertebral venous plexus located? This is the internal vertebral venous plexus located around the spinal cord, in the extradural space of the vertebral canal. And you see the fatty material here, the yellow fatty material, and embedded in these veins, these are the veins. So these longitudinal spinal veins, they drain into the internal vertebral venous plexus. Next. The internal vertebral venous plexus of anterior and the posterior all drain out through the intervertebral veins. They come out through each intervertebral canal. Then they communicate with the external vertebral venous plexus. And from there, they drain into each segmental region of the spinal cord. This internal vertebral venous plexus continues up and communicates with the dual venous sinuses. Principally, two dual venous sinuses. Who will name those two sinuses for me? What is the inferior feature of sinus? Another is the occipital sinus. And the same internal vertebral venous plexus shown here, here, and here. And the root of the stem. And so this is the venous drainage of this spinal cord. Tomorrow we are going to start with the chemical formations. We will start not straight away with the chemical formations. First we will start with the pathophysiology of cellular circulation. Pathophysiology of strokes. Then we will go into a little bit of other topics. Like how pharmacist occurs, how orthopedist occurs. And then we will go into stroke. Then we will go into all the manifestations of stroke. And then we will see how it is. We will not finish tomorrow. We will continue into Monday. Okay.


A 21-year-old bolus man is dead after a one-vehicle accident near Uppsalaup. The Morrison County Sheriff's Department says Cody Babineau's truck went off a road and rolled into a ditch. Babineau was ejected from the vehicle and pronounced dead at the scene. The crash was reported just before 6 this morning.


Even though criminal law interests everyone and it's something everybody will deal with at least once, a lot of people don't know very much about it. Here are a few basic concepts that might help. Beware though, this isn't legal advice, just a general explanation. Let's start with the two most basic elements of any crime. First, mens rea. Now, mens rea is not as it sounds some sort of manly death ray. Men's rea is Latin for guilty mind. To commit a crime, you'd have a particular state of mind when you act. Obvious, right? Let's use an example.  You have to act with a guilty mind at the exact time of the criminal act.  You were thinking about something else. An act as rea is just a fancy word for a criminal act. Can't commit a crime without action. But even if you want to commit a crime, you may not actually be successful. This is called an attempt. An attempt is a special crime that requires two things be present.  Second, an act. Something significant enough that the attempted is showing they really want to commit the crime, but not actually committing the entire crime. The world is full of criminals who didn't quite pull it off for all kinds of reasons. The difference in the criminal act is the important part. If you committed the whole act, it wouldn't be attempted robbery, it would just be robbery. For example, grabbing a purse but not actually taking it would be a big enough act to be an attempt. But touching a purse and passing while you think about possibly grabbing it would not be enough, even though you were thinking about it. So it's important not to jump the gun on stopping an attempt. Having the intent is still important. You can't commit an attempt because you're being reckless or not paying attention, even if you could commit the underlying crime that way. So while you can recklessly commit manslaughter, you can't recklessly attempt manslaughter. You can't recklessly attempt to commit any crime for that matter. It might be another crime but not attempt because you have to intend to commit a crime for an attempt to take place. And there you have it.