Okay, very good morning to you. It is Tuesday 15th of June and gonna run you through and get up to speed on the clothes on Wall Street What happened overnight in Asia and then some of the highlights for what's on the calendar for the session ahead and Overall this morning just looking across the charts here things are relatively quiet one thing that you probably can observe though is a bit of a late ramp here in Wall Street on the center charts here Which is the nasdaq 100 the S&P 500? And actually we've just added to that a little bit and then faded off those highs during the Asia pack session But these are all-time fresh record highs as far as the nasdaq the S&P is concerned on the actual print on the clothes continue to see really a digestion of that more rotational play Albeit it's moderate, but definitely evident in post CPI trade Which is what had been yields generally depressed and that's been helping large tech fangs were outperforming again yesterday And then as that was up around nine tenths of one percent compared to the Dow Which actually finished down around a quarter percent We've seen that pretty much consistent pattern of tech outperformance since that CPI figure came out on Thursday So the last couple of sessions One thing I would say is though is that things are relatively flat for the moment There's not really a great deal of real meaningful information overnight Although I do want to update you on a on a few things So maybe just a little bit of a light profit taking from those highs in Asia pack session But if you're looking at the S&P 500 here this morning, it is still sat at around those highs So you can see here and that was that late ramp going into the final half an hour of trade on Wall Street One thing to say is that that looks like perhaps a news catalyst has created that move But I wasn't at my desk at half eight last night But from what I can see on the headline feed it doesn't look like anything and very much often the case going into that final half an hour of trade If you ever look at the market on close the MOC If you're not familiar with that terminology if you just check it out on invest investopedia, for example But market on close can often see a fairly significant imbalance on say buy side or sell side And you do tend to see quite large fluctuation fluctuations going into the final half an hour of trade So perhaps that was a little bit of play and and now technically we've got that previous high And we've got a bit of a platform from where we hit that initial high Just after or at the closing bell and then pushed through it and used it as a support area during the Asia pack session So kind of confided by their 42 48 and a quarter in the R1 Which is just above the all-time high in Asia pack double top that we've traded so far around 42 54 55 at the moment Otherwise elsewhere One of the other things I guess to have a look at was gold Have seen gold come under some considerable pressure Yesterday and it was a bit of a continuation of the move that we're seeing at the end of last week popped higher Shortly through the Comics open and when the US started to really come into market and short term I'm just keeping an eye on Probably this kind of range at the moment of what we were trading In the Asia pack session before a brief move lower that we saw Again pretty light in the way of any real catalysts for that move But obviously liquidity fairly thin in the overnight session and as such then we've been responding to a short term trend line Marked from the low from yesterday when we were trading around the 1845 area and it's been Used a couple times and responded to On the Asia pack trading hours So around there increasing up towards through the pivot level here going into the european Open this morning So near term just having a look at that trend line pivot on the downside 64 and a half And then the upside a pack high at 69 for the time being any break up above there And then you've got 1870 kind of two a area which was that high seen yesterday afternoon And also the low print that we had Towards the back end of last week and any push on above that then be looking up to this area here of support and consolidation that we saw around 77 78 And gold though under a bit of pressure Of late overall and it has come as yields have seen a little bit of movement So the us 10 year you'll remember was just on a pretty one-way train Moving higher as yields continue to back off last week and obviously yields Broke through 1.5 percent. We actually saw a move to the lowest level since march But yesterday saw a bit of reversal of that Perhaps a bit of profit-taking on that fairly one-dimensional move that we saw last week It just faded a little bit and just having a look at us 10 year yields This is looking it's a bit of perspective here on on the chart and time frame This is going back over the last About two years or so and here you can see this is the move That we had during the onset of the pandemic This is looking at us 10 year yields And just wanted to really emphasize the importance of the area that we're trading at the moment, which is 1.5 You can see here that really was the low Of that move that we saw When yields were declining going back into the summer of 2019 came back to it towards the Autumn and then had another test of going back in early part of 2020 before then the eventual breakdown That ensued when yields collapsed on the onset of the the pandemic and the obviously response in monetary policy and fiscal policy and so on Now we've come back through that you can see again It acted as an area of resistance and now we dropped through it. We're sitting back on there So this is quite a key area around 1.5 in the us 10s and definitely as far as today's session is concerned I think it probably could be quite quiet now We're probably just want to wait for the fed meeting although Not expecting perhaps quite as much from that fed meeting as I was a couple weeks ago Seems like the can's going to be kicked a little bit before the real heated discussions around tapering But nonetheless, probably the 10 year might trade fairly Muted today as we await then more information coming out with the latest projections and a dot plot and so on in tomorrow's Session otherwise finally with the charts FX markets pretty quiet dollars pretty flat and if you look at the pound Just having a little break above an area of short-term resistance here at 141 21 As you can see as far as yesterday evening was concerned absolutely zero reaction to the extension of course of the Lockdown measures. This was very well telegraphed in the press and comes in absolutely no surprise So p.m. Johnson last night confirming the four-week extension to lockdown The government's expected to put the delay to debate And vote on wednesday in parliament And it is likely that johnson's going to face anger from conservative colleagues for backtracking and what they were dubbing as freedom day A couple of points that I mentioned in my morning notes on this and if you want to check those out You can find them on my my twitter account Or or in the amphi live community But my my point really being that irrespective of the political headache that johnson's government's likely to face From a number of internal conservative members I actually think that No, logically you would think well delaying reopening is going to be a net negative for sterling For example, because it's going to impede economic activity I actually think the opposite because a lot of the mathematical modeling that I was following from various different people who are who who have much more Intelligence than I were coming out with some pretty interesting and very frightening Estimates of what this wave could look like If we were to go ahead and reopen on the 21st Predominately based on lots of different reasons as well The ones that we talked about about still some remaining over 50s to get fully vaccinated Time for that vaccine to kick in in terms of its full level of immunity from it And then also as well the fact that we're still in the school year as well with the the transmission amongst the young This all coming of course because now Nine out of ten cases in the uk are the delta variant and so that being more transmissible than the alpha variant And that of then the original covid so much more Um ability to transmit and therefore for me I think that this actually this delay could end up being a net positive overall I don't I don't mean just by the pound this morning, but I definitely don't think it's a negative and actually I think it's a positive step Economically because although there's some short-term pain obviously from not fully reopening I actually think that you're mitigating the risk that what could have been much more stringent onerous restrictions that would have had to have come in If as some of the the scientific modeling was suggesting We could have seen a wave to the same proportion of that of what we had in the spring of 2020 on the original onset of covid 19 So I know this is a very contentious issue. There's also people that would say we should have reopened but for me The scientific evidence is pretty compelling and acting on that I think then safeguards us from having a more contained Latest wave which will inevitably allow us to open Sooner rather than what would have been later if we would have reopened It would have spread and we would have had to have gone into a firm lockdown thereafter So for me, this is actually a positive rather than a negative um I guess the other thing is that as much as there's Tory members that are going to Be really annoyed and frustrated by this and that's going to be evident in the Commons vote on Wednesday. I'm sure The latest you gov poll which came out last night after this was confirmed by Johnson's government Showed that 71 percent of English adults Supported the move to delay the reopening by four weeks. So the overwhelming majority So as far as that is concerned, I think Boris will be able to bat off any of that internal infighting Because the public seemingly agree with that move The government will review the date again on June 28th with the possibility of easing restrictions on July 5th If it looks better than expected But in honesty, this is considered as pretty unlikely at this point So it will go ahead as as per the statement last night But as I said pound hasn't reacted at all to this um in other news Biden continues to do the kind of round so to speak. So this time he's been at NATO um and NATO leaders Designated China that's presenting a systemic challenge in a summit communique on Monday Taking a forceful stance towards Beijing. It was biden's first summit with the alliance And beijing has responded and they've said that NATO needs to stop exaggerating the threat theory This of course comes after pretty similar rhetoric that we have from the g7 and cornwall at the weekend. So biden As we were kind of talking about doing his best to go to all these kind of unifying groups to try and bring together this collective stance against China at this point, which is something we knew that was high on the biden agenda with this latest tour President biden also stated that russia and china are looking to drive a wedge In the transatlantic solidarity And that we will respond if russia continues harm for activities While he added that russia and president putin chooses not to cooperate on cyber security and other things Then the us will respond and of course All of this is very much the optics around the fact that biden's going to be meeting putin tomorrow in switzerland So there's a lot of flexing of muscles You know kind of a posturing and these types of things again as far as the putin meeting is concerned We're not looking or expecting anything in terms of tangible results It's more about the atmosphere that surrounds the discussions that they have Does that then conclude their conversations in the joint statement? Which is deemed to be a positive rather than them to just log ahead to going their own ways And do they then commit to having further dialogue in the future? It's probably the best case scenario in that sense. So yeah, again, nothing really actionable Intra day, but something to be aware of No let up from the us on china at the moment There was a few other small things overnight, which probably just warrant a mention But again, it's more of an update to be aware of than something that meaningful The rba minutes From the june meeting stated the board agreed it would be premature to end Or consider ending the bond buying program and policy needs to remain highly accommodative in order to achieve full employment No real reaction seen in the osse overnight again. These are the dated minutes and then from a u.s. stimulus perspective Um, you know, is this thing ever going to get done on the infrastructure side? Well, the latest there is that senator manchin said further By partisan infrastructure talks are coming this week The group of senators will set a proposal out by the end of the week And obviously manchin being quite key because he's that particular Senator who's really on the fence of not is he going to support biden's agenda on this infrastructure bill? As seen as a real power player in these talks The other then is the us senate majority leader schumer who has said the senate will move forward with another infrastructure package via Reconciliation, even if it doesn't have bipartisan support So china puts a pressure on that they'll look to use reconciliation to get around it If so necessary, so I think that's more targeted at just trying to keep some momentum behind the dialogue at the moment So nothing really too much to comment on there for the time being As far as the calendar is concerned for today's session We've just had some uk data come out since i've been talking So just to get you up to speed the uk employment change 113 000 versus expected 150 The unemployment rate 4.7 percent in line with expectations the uk average earnings exponents 5.6 percent touch above the expected 5.3 so Not really too much of any type of immediate reaction here, but cable has managed technically now Let's get above this rectangle area I've got a previous support and resistance and you can see after the breakthrough It's come back to use that as a bit of a launch pad of support to now just Eek out a more progressive move up towards the r1 here So, yeah with that data nothing really too surprising If anything the average earnings number is a little bit higher than expected with inline unemployment. So perhaps some Some moderate wins in the sale to help this move higher and as I said, I actually think the delay is not a negative Even though it does have some short-term repercussion on economic activity I actually think it's a positive move and that's how I think the market will see it in in controlling the potential more severe Case outbreak due to the transmissibility of the delta variant otherwise Other than that and pretty quiet for the morning and then you've got a really jam-packed 130 data set Coming out and that's because you've got three real figures new or fed manufacturing US PPI and the latest US retail sales figures They're all coming out at 130. So whenever you see the calendar quite Packed at one single moment It does tend to add to then the kind of Mix signals that you get on the back of data points because obviously It's going to be multiple variables. They could be conflicting signals Probably the ones I think that the market will look at with a bit of clarity will be retail sales That has seen some flip-flopping as we've seen to kind of stimulus check impacts go in and out through the january march figures So it will be interested to see how that is performing PPI is of course On the more inflation side and then on the manufacturing side for the new york fed figure As well, but that's expected to remain fairly consistent at 22 from previous 24 So 130 is going to be interesting for sure I'd say that will likely act as then probably the main catalyst for Subsequent price movement for the rest of the session ahead of the open then on wall street an hour later 230 As far as then later in the session, there are a few more things industrial production from the u.s Coming out capital utilization at 215 You've got business inventories and na hb housing market index at three o'clock And then you've got all infantry data from the api usual much later in the evening at 9 30 if you're looking at it London time. So quite a jam-packed day actually after what was quite a quite calendar obviously last week and and yesterday to some respect So things should pick up pace on that front. Certainly Going forward now Speaker wise the chief economist from the ecb philip lane speaks at midday Bank of England governor bailey speaks at 115 and he speaks panetta just before 3 p.m The two ecb speakers will have no text according to the media office at the ecb and bank of england governor bailey is going to be speaking at a city a.m conference on financial services So i'm not expecting or anticipating any meaningful comments on either the current economic Conditions or future monetary policy, but nonetheless to be aware of here speaking uk supply uk german and then 24 billion dollars coming in the 20-year bond auction later on this evening at 6 p.m But that is it. So i'm going to leave you with that Again, if you're watching this on youtube Really appreciate it if you could like the video help it boost it So it goes out share it as many people as possible subscribe to the channel And any questions at all feel free to leave a comment absolutely happy to help have a great day ahead guys Take care and see you tomorrow


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Continuing with the topic introduced last time, name is a study of higher homotopy groups. Let us now introduce the so-called relative homotopy groups. This module 16 has been divided into two parts because it is slightly lengthy. The idea is that I would like to take you a little deeper here to the extent that more study of homotopy groups is not possible within our limits unless we use homology and homology theories. So at that door I am going to leave here. So let us do a little more of higher homotopy groups here. Given a topological pair XA with a base point X not belonging to A, sometimes I may forget to write down this base point that is all but it is always there in the mind, okay. Whenever you are discussing homotopy groups, pi 1, pi 2, etc., okay. Let us consider the following subspace of all paths in X, namely continuous functions from closed interval I to X. So this is the function space, it has, remember, it has a compact open topology. So now I am writing a subspace which I am going to denote by omega XA X not. Because omega is usually denoted for loops and pathies in spaces in algebraic topology. So all omega belong to X power I, that means a path from I to X, the starting point of omega is X not and the end point must be inside A, A is a subset, okay. So that is omega XA X not, it is a subspace of X power I. For n greater than or equal to 1, we will have this notation, pi n of XA which is a short form from pi n of XA X not, okay, it is a lazy notation that is all. The base point is always there, is equal to pi n minus 1 of this space omega XA X not, with base point for this one is a constant function at X not, a constant loop at X not. Where C X not is a constant path at X not. Take that, take the n minus 1 thermotomy set, for n equal to 1 this will be just a set if you remember that, for n greater than or equal to 2 this will be pi 1, pi 2 and so on they are all thermotomy groups, so that is just the definition, okay. For n greater than or equal to 2, pi n of XA is a group which is abelian if n is greater than or equal to 3, this is what we have seen last time, these are called the relative thermotomy groups of the pointed pair XA X not. Of course if X not is changed, you know how to relate them and so on, there is some relation but the groups will be different, that is why the base point has to be mentioned, for n equal to 1 in general it is not a group, we treat it as a set but a special kind of set, a set with a base point. The base point of the set itself is now a constant path, okay, it is a set with a base point whereas a group also has a base point and that is the trivial element of the group, okay, that is a pointed set, here is a group structure but pointed set that is there. So for n greater than or equal to 2, there are canonical bijections of the homotopy sets, pi n of XA X not can be thought of as homotopy classes of these triples, I n, I n is I cross I cross I n times boundary of I n and p is standard notation 0 0 0, okay, going into maps going into I n going into X, boundary of I n goes to A, p is always goes to X not and they are all corresponding homotopies, when you take homotopy class when you take I n cross boundary of I n cross p comma p cross I, so I n cross I is inside I, boundary of I n cross I is inside A, p cross whole of I must be a single point, that is the meaning of this notation. So this homotopy set is the same thing as pi n of X X not we have defined in a different way, remember that, pi n is defined as this one, okay, so when you defined like this it is automatically group structure, here you do not know, you have to still define group structures and so on that is a cumbersome method but this description is much simpler, okay, it is just like what you wanted, whatever you wanted, okay, only the second entry here are extra, otherwise it is just I n boundary of I n X comma X not, right, that would have been a definition of pi n, so pi n of X is this one extra, okay, so that is one, okay, there is one relation here that this is a canonical bijection, it is another one, pi n of X X not, it is also equal to complicated looking but this is a very useful thing, pay attention to this, I n boundary of I n and then instead of just single point 0 0 I am taking I n minus 1 cross 0 union boundary of I n minus 1 cross I n, all this is a subset of boundary of I n and it also includes p cross 0 0 0 is already there, this p point is also there, so it is larger set going into X not, so take only maps which take this larger set going into X not, boundary of I n go to A and I n go into X of course, only takes such maps and their homotopes, these two are equivalent and both of them are equivalent to pi n minus 1 of omega X A, that is the definition of this one, so this is a statement of theorem 1.18 which gives you three equivalent way of looking at pi n of X A, okay, let us prove this one, so let us have a temporary notation for this big set, third is I n minus 1 cross 0 boundary of I n minus 1 cross, there is a third subset here, so this I am renoting by z, okay. We know that this z included in the boundary of I n is a co-fibration because it is just a sub complex, okay, remember boundary of I n has this is a subset boundary of I n minus 1 cross I, union I n minus 1 cross 0 and I n minus 1 cross 1, that is the full boundary, so what is missing, only I n minus 1 cross 1, one of the phase is missing here, you take a cube, okay, take one of the phases and look at all the other phases, so that is what is happening here and that third is a included boundary of I n is a co-fibration, this we have proved in part one actually, not only that this third is contractible, so that is also easy to see, it is like a, on the sphere it is like a half half disk, half sphere, it is contractible. Consider the quotient map cube from boundary of I n to boundary of I n by z, we are z to boundary of I n is a co-fibration and it is contractible you are collapsing it, okay, therefore this is an equivalence, this is the amount of equivalence which we identify, okay, where z is to a single point z, okay, so that is the meaning of this quotient map, only all the points of z are equivalent to every other point of z, so that is the equivalence relation here, okay, so that is what we keep saying collapse that to a single point, from theorem 0.6, this is reference you can have from our part one course, okay, so I will just give you this reference here and show you, so this is what it is, so a to x is a co-fibration, it is contractible, then the quotient map is a homotopic equivalence, okay, so this was the theorem, all right, let us go back now, okay, so this is a homotopic equivalence, this quotient map, take lambda from boundary of I n by z to boundary of I n as a homotopic inverse of q, okay, so what is the meaning of that lambda composite q and q composite lambda are homotopic identity of the respective spaces, by composing it with a rotation of boundary of I n, boundary of I n is nothing but Sn minus 1, and Sn minus 1, you can have a rotation which takes any point to any other point, therefore you can assume that lambda of the single point z is the point p which is 0, 0, 0, lambda is a fixed homotopic inverse, you do not know where it z goes, if z goes here fine, otherwise you just rotate it and then you put lambda equal to this one, boundary of I n that, now you take put eta equal to lambda composite, okay, so lambda is, look at this one, lambda is boundary of z to boundary of I n and q is from boundary of I n to this one, so what you get, lambda composite q is from boundary of I n to boundary of I n itself and we know that it is homotopic equivalent to identity map, therefore it is homotopic equivalent, okay, so eta lambda composite q is a homotopic to the identity map and since the entire of z, z is a set under lambda it is going to a single point, under q the z goes to the single point z here, the whole set goes to the whole set, so under composite, so this eta sends the entire z to a single point and yet it is a homotopic equivalence from the boundary of I n to boundary of I n, this what is the point you have to understand. Now whenever you have a map from s n to s n, s n minus 1 to s n minus 1, you can extend it by cone construction to d n to d n, so that is what we do, that eta hat be the extension by taking cone construction I n boundary of I n to I n boundary of I n, automatically this will be also a homotopic equivalence, it is actually homotopic identity map, okay, it is stronger than saying homotopic equivalence and since the entire of z to a single point, now you already why I am doing all this, now you automatically have appreciated why I am doing this, this whole thing can be treated as a single point like this, that is the idea, okay, therefore eta hat induces a bijection eta hat prime, eta hat star from I n boundary of Y n p to x A x naught, suppose you have a map here into x A x naught, okay, then by composing with eta, eta hat, you will get a map from I n boundary of Y n as it is, this whole thing going to a single point p, okay, so omega goes to omega composite eta hat, that is the map from the first set to second set, these are all function spaces, oh, that is a bijection, automatically this composition is under, you know, under function space topologies, it is, if eta is continuous, composing this will be continuous, so this is a, this is not just a bijection, it is a homeomorphism, right now we are going to use it only as a bijection, under the exponential correspondence, the latter space is actually equal to this what you have to see, why we have taken this one, this is the same thing as maps from I n minus 1 boundary of I n to omega x A x naught, remember what is omega x A x naught, these are all paths in x, starting at x naught with end point in A, right, look at a function from I n minus 1 to here, under the exponential correspondence, it corresponds to map from I n to x, okay, so fixing a point x, you have a path x cross I, the starting point is x naught, but that is I n minus 1 cross 0 goes to x naught, the end point is inside A, so that is already, you know, whatever path cross I, that is going to be inside A, okay, so that is not a problem, boundary of I n also goes inside A, okay, more important is its entire thing going into boundary of I n, okay, boundary of, boundary of I n, on boundary of I n, the maps are here going to see x naught, it is a constant map, okay, therefore, this boundary of I n cross I goes into see x naught, okay, by the very definite boundary of I n here goes to single point, constant path, therefore this whole thing going to x naught here, okay, so this is the meaning of this set, this set is under, and the same thing as this set, okay, so this implies over seven, so why we have what this one, that is the seven, directly as sets themselves they are same, when you take homotopy classes, okay, if you have two spaces which are homeomorphic, then you take path components, we said nothing but homotopy classes are path components, path components are in one one correspondence, so it gives you seven, okay, now six is one step further, namely since eta hat is a homotopic to identity of you, upon taking path component we get six, okay, the functoriality of these bijections follows from the fact that everything is happening in the domain and has nothing to do with the topology pair x x naught, what is the meaning of this, suppose you have y comma b comma x, okay, a map from here to here composed with that map x a to y b will give you another map there, okay, now if you have another space, so z comma c comma that one, some function f and g and so on, all these things, composition is taking on one side, the map from here to here, this composition are changing of this from here to here etc, it tries taking on the domain, therefore they have because of associativity of the composition, the functoriality of these isomorphisms follows, okay, so this will be explained completely, not exactly this one, when you study the functors and categories and so on, okay, after that you come back and see whether it makes better sense for you, okay, so six gives us an alternative definition of pi n of x x naught, which is easy to remember and useful, however with this definition define the group structure on it will be somewhat difficult as compared to definition, the first definition 1.21, wherein the pi n's are automatically defined as groups, right, pi n minus 1 of some other space, so 7 is just an elaborate description of what is happening the function space x power i, but that is also useful, so you can keep all the three in your mind, okay, so that is just the definition so far, okay. Now take a special case when a is x naught, it follows that from this definition now 6 1, if a is x naught boundary of i n goes to single point, then p going to single point is redundant, okay, so it as shall write i n boundary of i n x comma x naught, right, so that is same thing as pi n of x x naught x naught, which is what which is just pi n of x naught, x x naught, okay, so that is the conclusion here, okay, so pi n of x a x naught is equal to pi n of x x naught x naught, it is pi n of x x naught, okay. We can now state and prove one of the most fundamental results in homotopy theory, given a pointed topological pair x a, we shall use the notation a included in x, x naught x naught is you know it is a pairs that is all, x x naught included in x a, so this is by j and this is by i, these in turn induce respective inclusion maps when you pass on to the loop spaces, the function spaces, omega a x naught instead of all path is sin a with certain property, right, starting at x naught, right, so you compose, you know if path is here compose with i, think of this inclusion map, it will it can think of this as a path in x, so it will give you omega x x naught, this inclusion map, similarly omega x x naught can be thought of as omega x a x naught, okay, so these are loop spaces, this is path and point could be any point inside a, okay, so these are inclusion maps which we shall denote again by i and j, all right, consider the evaluation map, you see x is a path, this x could x raised to i is a path from, this path space is from, right, from i to x, all the starting point is x naught, right, you can take that one, end point is important for us, look at the end point here, e 1 of omega, this evaluation omega 1, okay, so that is the map from x power i to x, now given alpha from i n minus 1 to omega x a, you take even composite alpha of i n minus 1 will be inside a, because that is the definition of omega x a, end points are inside a, therefore we have a homomorphism boundary I am denoting this one, it is induced by even composite alpha, okay, we are taking a simplified notation, this is classical notation anyway, pi n of x a, okay, remember this is homotopy classes of maps like alpha, okay, 2, you get a map from i n minus 1 to a, so that will represent pi n minus 1 of i n, so that definition of boundary of alpha is nothing but the class of even composite alpha, okay, using definition says it is clear that boundary of alpha is nothing but just alpha restricted to boundary of i n, the last coordinate restricted to boundary of i n on the boundary, okay, if you think of this as maps from here, maps from i n minus 1 cross i, that is the i n, so boundary of i n is fine, okay. For any pointed topological pair, there exists a long exact sequence of groups and homomorphism induced by this topological pair using the i, j and daba, so all these three will be occurring again and again and they will be forming a chain, infinite chain of homotopy groups and this is the homotopy groups. Pi n of a, the first one is inclusion map, i check, inclusion induced, then j check, inclusion induced, then boundary, here for the sake of accommodating the entire thing in a single line, I have omitted mentioning the base points, the base points are there, okay, always. These are all homotopy groups for n greater than or equal to 2, so these are groups and homomorphisms, what is the meaning of exact sequence, any two compositions are 0, that is funny but it is stronger than that, kernel of j check is equal to image of i check, kernel of daba is equal to image of j check, kernel of i check is equal to image of daba check, this should happen at every group here, goes on from for n greater than or equal to 2, they are homomorphisms and keep coming down, down n minus 1, n minus 2 and so on, last thing will be pi n of x, pi n of x a, pi not of a, pi not of x, this is a group, okay, but this is not a group, this is not a group, this is not a group, so what is the meaning of exactness here, remember that is why, remember these sets are pointed sets, so there is a constant function here which is the base point, okay, so constant function here again base point here, these also base point constant function, so constant function works fine, everything which goes to constant function as kernel must be the entire image of daba, similarly everything which goes to the constant function under daba must be entire image of j check, that is the meaning of the, this sequence is exact, okay, kernel and image can be defined when you have pointed sets, so let us prove this, obviously the proof will be divided into three parts, namely exactness set here, exactness here, exactness here, okay, so one by one, the proof is broken into three steps, the first step is kernel of j check is equal to image of i check, kernel of j check here, the i check will come from here, this one, j check and i check, so pi 1 of x, okay, so start with an element alpha in pi n of x which is by definition pi n minus 1 of omega x naught represented by a map i a alpha from i n minus 1 to omega x x naught, where this alpha itself is i composite beta for some beta from i n minus 1 to omega x a, that is you are taking an element here which is the image of i check, alpha is i check of beta, I have to show that, j check of that is 0, that will show that the image is contained in the kernel, okay, that is one part, the second part will be that kernel is contained in the image, okay, so let us prove that image is contained in the kernel, so I have picked up this alpha, so set alpha is i composite beta, we have to show that alpha is in kernel of j, just define a homotopy h of x t s equal to beta s t s x s, s is a s is a real number between 0 and 1, okay, s and t, x is an element of this i n minus 1 or i n wherever, okay, the beta of s x makes sense, okay, beta of s x is a path operated upon t, so this h is a map from i n minus 1, the first slot is x cross i second slot is t, then it is a path in x, so it operates upon again an element 0 to 1, okay, so gives you a path in omega x a, why it is a path in omega x a, put s equal to 0, what is this, it is beta 0 of t, the beta 0 of t is always a constant function, so that is a point x naught, put s equal to 1, it is beta x of t, beta is taking values inside a, therefore the end points are inside a, so that is the meaning of this, so this is the way you have to verify, okay, so verify that h from i n minus 1 cross i to omega x a is a homotopy of the constant loop at x naught with alpha, okay, so if alpha is homotopy of the constant loop and what is happening, only when you go to omega x a, therefore alpha is in the kernel of j check, okay, when you go to omega x a 0, so one part is over, the second and the converse part given alpha and pi n of x a, so put j check of alpha is 1, this gives us a homotopy, j check of alpha is 1 means this alpha when you treat it as an element of omega x a is an homotopy, so take the homotopy h from i n minus 1 cross i to omega x a from alpha is a constant, okay. Now take g as i n minus 1 cross i cross i to x given by, I am reversing the rules of what we did in the previous step, g of x t of s is but it will carefully have to h of x t s 1 minus t times s plus t, okay, s as a general factor here and the constant function 1, constant the point 1 has been taken 1 minus t plus 1 minus t of this plus t times that is converse combination of that, you have to verify that g defines homotopy of alpha with a map beta given by n point namely beta x of s is nothing but h of x of h x 1 of 1, okay, when this whole thing is 1 that is the homotopy of the n point, then when it operates on 0 that will be alpha that is what you have to verify, this means that beta is an element of pi n minus 1 of omega x a omega a, okay. Therefore, if alpha is homotopy to i composite beta it is same thing as i check of beta itself, so we have to prove two more steps here, kernel of dab i equal to image of j check, okay, so now we are coming here this part, kernel here is equal to image of j check, elements are inside pi n of x a, okay, so if alpha is j composite beta for some beta so i n minus 1 of i n minus 1 into omega f, okay, then even composite alpha remember this was different this was used in the definition of the dabba the boundary operator, even composite alpha which is the end point map is constant map c x naught because beta is a loop at x naught, this means dabba of alpha is just a 1 that means the trivial element is c x naught, okay, so one way is very easy, now we have to converse, suppose dabba alpha is 1 for some alpha from i n minus 1 to omega x, this means that we have a null homotopy of even composite alpha, okay, you see even composite alpha may not be a constant function but it is homotopic to constant function is what we get, okay, it is relative to the boundary, okay, that is the meaning of that this is a trivial element in this group, okay, i n minus 1 to a boundary of i n minus 1 going to single point all the time that is the meaning of this group, alright, so what we have we have a null homotopy of even composite alpha from i n minus 1 to a, okay, let h be such a homotopy, for each fixed x in i n minus 1 let us define a path omega x from 0 1 to a, okay, omega x is a path operating upon s is h of x, this homotopy, then omega x is a path from starting from even of alpha x to alpha x of 1, okay, which is 2 x naught inside a, okay, so even of alpha x is not exactly all the time x naught, right, so this is alpha x of 1 by definition and it is homotopic to constant power so 2 x naught you get, each of them will be end point would be just x naught, so alpha x 1 is some point and from there I am joining it to x naught that is the meaning of this one, omega x is what that is what is doing, so now put omega x t of s equal to this omega x t s, okay, so I am defining another path here, okay, tempered path, now you define h from i n minus 1 cross i to omega x by h of x t equal to first part is alpha x, remember this is a path it ends in alpha x 1, so take that point and take omega x t from there, so omega x t starts at alpha 1, alpha x 1 and ends at some other point, okay, so take this path this, okay, I want to say that they are all loops now, what is the end point of this is always x naught, so starting point is x naught, end point is x naught, so they are loops, so we have h of x 0 is alpha x star c x, okay, alpha x star c x naught, what is the point is x is always alpha x, so I have to put 0 here, so alpha x 0 is just c x naught that is the same thing as if you take the constant loop and take alpha x constant loop, constant path it is always alpha x, right, homotopypt alpha x and h of x 1 is alpha x star omega x t equal to 1 it is omega x, remember what is omega x t of this one, omega x 0 will be omega x of 0 comma s whatever s is it is really 0, omega x 1 will be omega x of just s it is omega x, okay, so alpha x star, alpha x star omega x is h of x 1, okay, this we know is an element of omega x that is why we started with this one, okay, this is how we started with this omega x is this one, okay, they are all loops, so this we get a homotopy of alpha with a map beta which is h of x comma 1 at end point, that map is i n minus 1 omega x, this alpha end point was arbitrary point inside a from there we took path is going into x naught, so all of them are now loops and these are homotopy, therefore this alpha is in the image of J check now, so this alpha being homotopypt beta it is image of J check, so this is a clever proof it needed some cleverness here, okay, the third step kernel of i check is equal to image of Davao, this last thing here kernel of i check is image of Davao, so now elements are inside pi n minus 1 of a, okay, let now beta from i n minus 1 s n minus 1 to a x naught be a map such that beta is in the image of Davao then I have to show that i check of that is 0, i check of that is constant function that is what I have shown, right, so this implies that there is a map alpha from i n to omega x a such that this beta x is alpha x 1 that is the definition of that beta belongs to image of image of Davao, okay, putting h of x t equal to alpha of x t remember alpha x is a function, so h of x t becomes homotopy, we have h of x naught is equal to alpha x at 0 x x naught h of x 1 is alpha x 1 is beta x, hence beta is null homotopy the starting point is a constant function and this beta, so beta is null homotopy in x, okay, so that means beta is in the kernel of i check, when you go to i check i is from a to x, right, okay, so one way we have proved conversely suppose you have a homotopy from i n cross i to x, okay, to a constant loop at x naught with the map beta because beta is null homotopy in x that is the meaning, then look at alpha from i n to omega x a given by alpha x of t equal to h of x t the same homotopy, so that gives you boundary of alpha is equal to beta because boundary of alpha is evaluation at the end point, the end point is beta starting point is x naught, so that will be a looping that is really path in omega x a that those are functions of omega x, that is a function from i n to omega x a, beta has property boundary of alpha is beta, so that proves beta is integer, okay, so to be precise 3 and 1, step 1 were easier, step 2 was somewhat difficult but in nature they are all slightly different that is why I have taken pain to write down all these proves, okay, so go through them carefully it is not that if you prove that one other things are similar this is not the case here, okay, so go through them carefully that is the only way to understand the various interrelations here and the definitions and so on, you know properly, okay, so we will take a rest here and start the second part of the module 16 a little later, thank you.


What is green screen? Why would you want to use it? We've all seen the weather person give a report with the giant map behind. How is that done?  At LPM we have installed a 12 foot wide by 12 foot long green screen. It's green paint on canvas. When you do a shoot in front of a green screen you're able to emulate not only the weatherman effect but you can place any scene behind. Whether it's video or a graphic or a photograph you can easily insert anything into that green area and have your talent look natural in front. Right here we have the controller. As you see there are two rocker switches up and down. One is for the paper backdrop and the other is for the green screen. Now you want to be careful when you roll the green screen down all the way to the floor that you don't let it bend and wrinkle. You can easily just pick it up from the middle and lay it back on the floor so it's nice and smooth and you get a nice little cove. When you roll it up though you want to make sure that you don't roll it up too high otherwise it could possibly get stuck. It's very important that the green screen is evenly lit. The way you can ensure that it's evenly lit is by using a waveform monitor. On the waveform monitor what you really want to see is a straight line that represents the full width of the green screen. The two lights that are used to throw even light across the green screen are the Godox 200s. We set those at 5600k and that emulates daylight. Everything on the set, all the lights should be set to 5600k as well as the camera. Now you want to have your talent at least four or five feet in front of the green screen. It's advisable to use the big soft boxes, the large ellen chrome soft boxes and what they do is they cast a soft light onto the talent so you will not get harsh shadows. It's probably going to take a little bit of experimentation to find the right level to set the lights at but I would start at about 60%. Do not ever step on the canvas green screen and most importantly talent should not be wearing light green or green of any type that matches the green screen. Here's why you don't want your talent to wear a green shirt.


Siberian cat vs Sphinx cat. What are the differences between these two cat breeds? The Siberian has a long triple coat of guard hair, an outer coat, on hair, the middle part of the coat, and a downy undercoat.  In cold climates, the undercoat thickens. The coat is available in all colors and color combinations, with or without white. He looks powerful and alert, but he looks at the world with a sweet expression. His head is a modified wedge with rounded contours, brought at the top and slightly compressed at the muzzle. Medium large ears are well equipped with hair tufts. The almost round eyes can be green, golden, green gold, or copper. White Siberians or Siberians with white patches can have blue or odd eyes. His body is muscular and he is big round paws with tufts of fur and a thickly fur tail. Bald and wrinkled, the Sphinx is often proclaimed ugly, but usually by those who have a shallow understanding of beauty. His unique body form and the physiological and emotional warmth he emits are what attract people to him. Like an apricot, the hard, muscular body of the Sphinx may be smooth or coated with a fine, soft down. The nose, the toes, the ears, and the tail can all have a light fur coat. To embrace him is to disappear into the comfort of his suede-like coat. Sphinx does not have a higher body temperature than most cats, but it seems like it is because there is no wool to be used as insulation. The head of the Sphinx is a modified wedge shape, slightly longer than the wide one, with high cheekbones, large ears, and large, lemon-shaped eyes. Whiskers and eyebrows are sparse to nonexistent. The Sphinx comes in all colors and patterns, including white, black, red, chocolate, lavender, various tabby patterns, tortoise shell, calico, bicolor, and pointed and mink patterns. The color is seen in the pigment of the skin as well as in whatever hair the cat has and can sometimes be difficult to distinguish. The heart of the Siberian is as warm as his homeland's cold. He loves people and wants to be next to them, so expect this affectionate cat to follow you around, even in the bathroom, and to support you with all your reading, TV viewing, computer work, and meal preparation. Sitting in your lap as you comb their fur may well be the highlight of your day. When you get home from work, he may not have a martini waiting, but he'll be happy to tell you all about his day in calm, pleasant trills and chirps, interspersed with a few meows and purrs. Guests will find him a brilliant host, he's not typically a cat who's shy in the company of strangers. Besides being loving and attentive, Siberian is active and playful as well. He's going to instigate games of fetch by getting you a favorite toy to toss. Any object may be a plaything for the smart cat, so keep jewelry or other potentially intriguing objects out of his sight. Teaching him tricks is a fun, easy way to challenge his agile brain. He loves to play with water because of his heritage as a forest dweller, maybe it's a genetic recollection of going fishing for his supper. Don't be surprised if he enjoys splashing you in a bathtub, drinking from a fountain or a faucet, or making puddles with his paw in his water bowl. As for a working and hunting cat, he is extremely agile, and you may find him balancing over a doorway, or propelling to the highest point in the room. Indeed, Siberian coaches may well be a secret to the success of Russian gymnasts. Activity and noise won't bother him a little bit. His relaxed nature offers him the ability to be a cat for therapy. For nothing else, he's going to be happy to snuggle with you when you're sick with a cough or some illness. And even though he loves affection, he's not in need, and he waits patiently for you to have time to commit to him. It's a good thing that the Sphinx loves attention because he's attracting it everywhere he's going. He's demanding human love and he's going to do anything for a laugh. Look at me. Is the cat phrase. That makes it easier for veterinarians or anybody else to deal with, and it's not uncommon for the Sphinx to be a therapy cat because he's so fond of meeting people. When he doesn't get the attention of his adoring fans, the adventurous and enthusiastic Sphinx explores his surroundings, climbs his cat tree, or otherwise tries high places, chases a bug, or otherwise gets into mischief. He is extremely fond of teaser toys and puzzles toys that challenges athleticism and minds. This is a highly social cat. If you are gone during the day, the Sphinx will enjoy having a friend in the form of another Sphinx, another cat, or even a dog. He likes the company and, equally importantly, he likes having someone to snuggle with so that he stays warm. He loves to be held, and you can expect him to sleep with you, most likely under the covers? Think of it as having a living hot water bottle. Hello. This video is sponsored by Beemix Pets. Are you looking for high-quality cat collars at an affordable cost? Check out BeemixPets.com. Use coupon code Kittenlife to get 20% off. Siberian's thick triple coat should be combed or brushed several times a week to avoid tangles or mats. The coat will be shed seasonally in spring and fall, and you will need to groom more often during that period. A bath is rarely required, which is a good thing since the coat is really waterproof. It can be hard to get a Siberian wet enough to shampoo him. Most of the Siberian development occurs in his first year and a half of his life. Your kitten breeder can suggest that you feed him the kitten food at that period to make sure he gets enough nutrients. A Sphinx, despite its bald body, requires at least as much grooming as for cats, and maybe even more. Their skin must be kept moisturized with a gentle, odorless lotion or oil, and they require a weekly bath so as not to leave greasy spots on your furniture and clothes. Use a gentle baby shampoo or a moisturizing shampoo and rinse thoroughly, particularly between the wrinkles. If you start bathing your Sphinx kitten when he's young, he'll learn to take a bath and maybe even enjoy a bath. Baby wipes are going to help keep him clean between the baths. Although the Sphinx feels warm to the touch, it doesn't have a fur coat to keep it warm. If you're cold, he's probably cold, too. Buy him a nice sweater or two to help keep him warm. If your Sphinx has an outside enclosure where you can sunbathe, make sure to add a cat-safe sunscreen to their skin to avoid sunburn. Brush the teeth to avoid periodontal disease. Regular oral hygiene is the best option, but weekly brushing is better than nothing. Wipe the corners of their eyes daily with a soft, damp cloth to remove any discharge. Use a separate area of the cloth for each eye so that you do not run the risk of transmitting any infection. Check their ears weekly. If they look dirty, wipe them with a cotton ball or a soft, wet cloth moisten with a combination of 50-50 cider vinegar and warm water. Avoid using cotton swabs that can damage the inside of the ear. Keep the litter box spotlessly clean. Like other cats, they are very particular about the hygiene of the bathroom. It's a smart idea to keep the Siberian and Sphinx as an indoor only pet to protect them from viruses transmitted by other animals, attacks by dogs or coyotes, and other threats posed by outdoor cats, such as being struck by a vehicle. Cats who go outside are often at risk of being stolen by someone who would like to see such a beautiful pet without paying for it. The Siberian has a bold temperament, and there's nothing much with his composure. These characteristics make it an ideal choice for a family with kids. No creatures of the night will get past the Siberian guard at the foot of the child's bunk. He's able to live with other cats and cat-friendly pets, too, as long as they know he's in charge. Responsive and social Sphinx is the ideal choice for households with children and cat-friendly dogs. He learns tricks quickly and enjoys the praise he gets from children who regard him kindly and respectfully. He lives in peace with dogs and other cats. Always integrate pets slowly and in controlled circumstances to ensure that they learn how to get along. If you enjoyed this video, kindly press the like button. Also don't forget to subscribe with notifications on, so that you don't miss out on videos like this. Thank you for watching.


update. Good afternoon, folks. Steve Rhodes coming to you live from the shores of sunny Delray Beach, Florida. This is your 1 p.m. update and a slightly mixed bag out here that mix coming from the semis are off about three points out there. So basically flat. The other new seas are not flat. The Dow's up two points, two percentage points that is. It's up 606 bucks. The S&P's up 1.6 percent. $62 there. One and two tenths for the Nasdaq 100, 143 to the upside. The Russell 2000 up 17 or 1 percent. The Wilshire is up 563 points. One and a half percent. The gold is up three tenths. Six bucks. Giving up most of its gains. Silver's up five cents. 21.72 is the print. Let's recruit up a buck. One 13.23 is the print there. And the third of your treasury is back a point and a half. Trade out at 140.10. As we begin the market update, we'll look at our nine panel market update charts out here. We've got the EES mini up a left hand corner. We tested and rejected the May 12 swing point on Friday. Price is back inside. It's a daily profile. It suggests well resistance is really going to be the next resistance level for it is the center of that profile. That's at 39.89. You close above that. You should get back to the highs from last week. And perhaps you get up to the top of that profile at 41.68. To get back to that level 41.68. That is you're going to need to see a spot while it is closed below its 50 day exponential. It's set a bit lower. Support there is at the 27.45 level. In the case of the NQ, the NQ testing and rejecting its swing point from May 12 last week. It's really running into resistance. You don't see it on this chart here at its asset or in change line. We'll look at that during the trader's edge show. The U.S. dollar index is pulled back to the support of a bullish structured weekly profile. So that says that what that level, which is 102.13. We're trading 102.17 right now is a very key level of weekly support. That means the close on Friday. Gold is forming a brand new weekly bear structured profile. Supported 1828. Resistance between 1937 and 1959. But the first level resistance is the daily timeframe. That's also a bear structured profile. And that's between 1861 and 1872. Silver also forming a new weekly profile. It's trading inside a bullish structured daily profile. So the resistance here that Silver is trying to overcome is 2226. Lights recruited just trading between trend line support, trend line resistance, 30 year treasury, I'm sorry, 30 year treasury is just consolidating in essence with inside its daily profile. But you can also see trend line support and trend line resistance there. Natural gas looks like once we'll start with that NQ. For the trader's edge show, if you'd like to start your money to have a magnificent one, thanks for joining us and we'll look forward to seeing you get soon.


Hello everyone, I'm Hela currently pursuing a master's degree in international human resource management at the University of Edinburgh Business School. As a dedicated student of modern HR practices, this event was a must-attent for me. We delved into vital topics such as AI integration in HR and evolving workforce skills through fireside chats and interactive sessions. The skills-focused sessions were particularly eye-opening, shedding light on the rapidly changing landscape of talent management. But perhaps one of the most valuable aspects was the networking opportunities. I connected with seasoned industry professionals and fellow students, forging meaningful connections and gaining invaluable insights. Thank you!