bounded below away from zero and there are two possible situations where either the curvature stays bounded along the floor, either the curvature is unbounded. I want to discuss today the two situations and what you can say. So the first situation I want to discuss is the fact. So they collapse with bounded curvature and already yesterday I told you that this can happen only when a single curve, isolated curve, collapse down because actually if you have a collapse with bounded curvature you can imagine that you cannot have a collapse of a region because there is some curvature where a region can collapse only if it has less than six edges, five edges and if you consider a region with five edges, we have five at most, five angles of 120 degrees and if you consider a straight segment between these angles, these corners, actually there is not enough curvature to close. So with only five angles of 120 degrees you need some curvature in your region in order to close. With six you can do it, you can have a straight hexagon but less than six. So your region in order to collapse has to have at most five edges so there is some curvature around and you can imagine then when a region collapse the curvature must go to plus infinity, it cannot stay bounded. This is absolutely a non-elementary lemma in general and actually in general we don't have a proof of this outside the word of moving networks. If you find the proof of this, a bound from below on the area for regions with five angles and of 120 degrees and the curvature bounded above, I would be very interested to know it, a proof of this but well in the case of networks you are in a very special situation, you can prove that. So if a region is collapsing the curvature is going to plus infinity so it's the other case that I'm going to discuss. So in this case a region cannot collapse and by the argument that I showed you yesterday that means that only an isolated single curve of your network is collapsing down not other cross curves which doesn't mean that only a single curve it could be that at same time in different points in different areas of your network some curves are collapsing at the same time but locally only one single curve and so the analysis is understanding what you get when you send your time t to big t and what you get is a limit some kind of limit network in order to reapply possibly one of the theorems or possibly one of the short time existence theorems that we saw at the very first lecture in particular the theorem the very general theorem of Ilman and Neves and Schulze in order to restart your flow. Okay now what happens is that actually possibly parametrizing all the curves what is it is possible to show proportional to arc length actually it's possible to show that the bound on the curve which implies some convergence in double two infinity and in c one to some limit network s big t as t goes to big t let me and the very important thing is that this convergence gives you it's not up to a sequence all the sequence of networks converge to a unique limit network actually and moreover even if the I'm not saying well sorry this is great so sorry I made a mistake here so the w two infinity convergence is weak while the w the c one converges is strong so the convergence is strong in c one and weak in w two infinity to this limit network so you only have that well you still have the the curves are of class w two infinity so not necessarily c two for sight but at least they have bounded curvature in a distribution from a distribution point of view. Moreover what happens is actually that s t is non-regular since a single curve is vanishing you can imagine that the situation is that two triple junctions are going to collide along this triple curve this this vanishing curve and what you get in the end since only this curve is vanishing by the argument of yesterday what you get at the end is a four point before showing some something related to this let me try to convince you that give you another idea of estimates that you need different by the previous ones actually several of the things that I said easily hide a lot of estimates behind that actually just give you a taste of the another estimate in order to get this so as you can remember we all we always working with this which means that lambda is equal to and this explicit expression of a tangential part of the velocity of the motion actually allows to write down an evolution equation for lambda which is the following this lambda evolves like t derivative is equal to lambda s s minus lambda s minus two k k s plus lambda k square and let me recall also the evolution equation for k which is k s s plus k s lambda plus k q so I want to convince you that actually you have a unique limit so the what follows by this guy but these two equations that if you consider this guy here and call it v it's the velocity at every point of the motion I want to write down an equation for the square of the velocity actually the velocity since these two guys are or to normal it's the square of the velocity it's simply the square of k plus the square of lambda and now I use these two when I derive this I get two lambda times all this stuff let me write this explicitly lambda s s minus two lambda square lambda s minus two lambda k k s plus lambda square k square then I derive k square sorry here is a four and here is two now I derive k square so what I have is plus two k k s s plus two k k s lambda plus k to the fourth power two and now you see that there are similar terms around this one is like this one so I can throw away this and put minus two here and now I want to rewrite all this guy in terms of v actually so what I can I deal with these two together these are equal to lambda square plus k square s s minus two times lambda lambda s square minus two times k s square I simply taking derivatives and okay now I'm looking at I use this guy here minus two and this guy here two together if I collect lambda this is equal to minus two lambda lambda lambda s plus k k s which is equal to minus two well minus lambda lambda square plus k square s and finally if I take one and two I collect the k square that is plus two times k square lambda square plus k square now you see that our v v square is equal to lambda square plus k square appears here minus lambda v square s plus two k square v square minus squares so throwing away these two I can write smaller or equal than this guy here so I wrote down a partial differential inequality for v square that actually works very well for maximum principle if you can use maximum principle because at the maximum point of v square this guy is non-positive this guy is zero if the maximum of v square is in the interior and what you find out that simply the time derivative of the maximum of v square is smaller than two times k square v square max and in our case the curvature is bounded so this guy here is a constant it can be bounded by a constant so this is bounded by a constant times v square max which means that if you can use maximum principle on this equation v square can only have exponential growth in particular infinite time cannot go to plus infinity you can bound the v square at some time at every positive time by its value at time zero and if you bound v square you are actually able to bound lambda square so a bound from k on k gives you a bound on lambda also in particular on the velocity of the motion all of these if you are if you can use maximum principle on this equation which means if the maximum of v square is in the interior not at the triple junction okay but at what happens at the triple junction at the triple junction we know from very beginning computation computation at the beginning that the sum of the curvature square are equal are exactly equal to the sum of the lambda square at every triple junction and now you know again that your curvature is bounded which means that at the triple junction all also the tangential velocity tangential part of velocity is bounded so at the triple junction you have your estimate that under the assumption that the curvature is bounded also the tangential part of velocity is bounded by some constant which is more or less the same that bounds the curvature so either during the flow inside the curve the tangential part of the velocity is still bounded by this constant the same constant which bounds the tangential velocity of the boundary either if it is larger than that then the maximum must get must fall in the inside and then you can use maximum principle because when you have this you also have that v square at the triple junction is bounded by two times the maximum of the curve well by some constant the maximum of the curvature square which is actually bounded by another constant by hypothesis so this either v square is bounded but its value at the boundary of the curve which is already bound uniformly bounded otherwise you use maximum principle in the interior on this equation to say that it cannot grow too much and so you get a uniform estimate on v and then also on lambda so bounds on the curvature implies bounds also on the tangential velocity which is very good let me underline only for this flow for this special flow that actually it's we can always put our self in this situation so we have this bound on the velocity and then I look at these guys I take one curve and the curve one point on this curve gamma i of x and t and I look at the difference between gamma i of x and s so I look at the square different times this curve the this point is mapped by the curve and this difference is actually equal to the integral between s and t let's take t larger than s s and t of delta gamma delta t and x xy this is simply deriving integrating but we know that this guy here is exactly the velocity of the motion it's exactly our v by the equation over there so this is equal let me do carry the modulus inside of the integral so this is equal then the integral between s and t of the modulus of the integrand which is actually the modulus of v at x c the xy but we we just derived the bound on v depending on the curve but sure so this is smaller than a constant times t minus s and the constant is independent of time actually we did it for the generic x and x doesn't appear here so actually this is you can also write that the uniform norm between the same curve at two different times is bounded by a constant times the difference in time so then it's easy to see that this implies the curve the map gamma i indexed by the time for time going to big t it's a Cauchy sequence in the c0 of zero one as t goes to t so it's a so you have a limit a unique limit actually of the curve gamma i that we can can call gamma big t from zero one to r2 which is our candidate limit network unique there's only one limit because of this fact that it's a Cauchy sequence in time indexed by time but this is only continuous map but then the parameterizing arc length and using the bound and the curvature what you conclude that possibly not excited this map but the what you see in the limit the support of the network in the limit it's actually well it's actually something which is parameterized in c1 with bounded with bounded curvature and it can be unique the limit is unique there is only one limit network in doing in doing this this convergence so we have a limit network at big t uniquely network it's curve this emotionally bounded curvature actually with some effort you can prove that actually far from from the possible multiple junction the curves are not only w2 infinity they are actually c infinity and far from the triple junction the convergence is also c infinity that's kind of regularity t or m that follows by several estimates for instance by interior estimates of hacker whisker so actually what you conclude that you get some unique limit big s big t one and w2 infinity weak to some s big t which is actually a c2 because if you are c2 with bounded second derivative outside the point then you are c2 up to the point actually and what's happening here that s of t from the geometric consideration is something like this with this curve is going to collapse so when t goes to big t what you see these other curves are not collapsing they are all converging c1 actually that you find the four point and actually you can expect that could happen something like this for instance 40 and 80 120 since the c1 convergence forget this part these two curves must converge to something that forms an angle of 120 degrees like here and also the other one so other two here this a priori could be possible but actually you can exclude this possibility where these two angles are differently by 60 degrees because in a way in order to do this there must be some curvature here on this curve that can is in a way concentrating and concentrating at this point making in a way some kind of turn in the in the collapsing but concentration of curvature means that the curvature is concentrating in a small length of this curve but going to plus infinity but here it is not a situation here we assume that curvature is bounded so actually what can be proved is that when you when this curve collapse what you see at the end are two curves like that with 120 degrees between them and exactly 60 degrees in the other angles so a single situation so all this stuff force to have a limit s t network forming a four point here actually not any kind of four point exactly with these angles 120 120 60 which is exactly what we saw in the simulation possibly i didn't underline too much the very first day that i show you the simulation but when there is a collapse of a single curve the angles behave like this actually we can re-see the simulation maybe tomorrow at the end to show you again the the the video so this is a theorem under multiplicity one conjecture that we use several times what we get in s t are it's a a network with triple junction triple junction where nothing happens stays triple junction regular triple junction with 120 degrees and some for every isolated curve like this which collapse you get one of these four points with this angle condition and as i said it's exactly what we saw in the simulation this is a better drawing i'm not very good at 10th but this is better thanks to alessandro plude okay just a remark this analysis can be extended also if you one boundary curve boundary arriving at an end point of your network if this curve is collapsing actually instead of an inner curve what you get in the limit it's two curves arriving at the point p forming between them an angle again of 120 degrees which is exactly what we saw when i show you the example of the spoon and the curve connecting the spoon to the boundary collapses and you remember if you remember in the end we got something like this with these two angles all 120 degrees with this angle of 120 degrees this is exactly an example of this situation of the boundary and actually if you get this depending on the what kind of physical or problem you are trying to modelize you have to ask yourself what you want to do with this guy when a curve on the boundary vanish actually how to restart these guys well actually all the theorems that we developed worked only as a underlying only for regular networks with triple junction so if you get a four-point you absolutely need the short time existence of ilman and neves and schultz in order to which is a more geometric major theory based okay now i want to consider very special networks three like so there are no loops around that's simply graph so you don't have any closest loop and what can be shown is that if you work if you consider net which is a tree and the one of the simplest one like this four point on the boundary and two triple junction is one of the simplest well actually what what can be proved is that if you look at this guy well it can happen that this curve we are collapsed and we are in the same situation here and this is in a way some kind of converse of the previous theorem if you are a tree well whatever the curvature cannot go cannot be unbounded actually the curvature is always bounded during the flow must then if you know that if your curvature is bounded then you are in the first situation so the only possibility of singularity is a single isolated curve collapsing down and doing again exactly this phenomenon here so it's a kind of if and only if if you are dealing with it if if you are dealing with a tree actually this is the only possible singularity is that you can that you can you can see during the flow okay before discussing some consequences just to give you a hint why this works well actually the idea is similar to what I just mentioned yesterday about the evolution of a single single tree actually that in a way if you control the curvature far not so far but the triple junction actually you are able to control not the curvature but the growth of the curvature close to the triple junction sometimes you control so good if a tree of this straight enough flat enough that they cannot go to plus infinity fast enough in order to produce a singularity actually the proof involves several estimates but the idea is that if you consider this guy a tree and you take as usual choose a point x naught as before and take a blow up with the standard rescaling dynamic whisking and dynamical rescaling take a blow up and get the limit shrinker as before of a tree well taking a blow up is like taking a limit and you can think you can guess that if you take a limit of something which is a tree you still have a tree because you cannot produce new regions you cannot produce loops also because of the multiplicity one conjecture forbids the possibility that you the network touch itself and produce produce a closed loop so limit of trees is a tree is a tree so the limit shrinker that comes out from a blow up in the in the situation of a tree is a tree and then you can work out a lemma that tells you that the shrinker which is a tree must be actually only it's an absolutely non-elementary lemma that if a shrinker is a tree then it must be composed only alfines that actually as we said to yesterday since an alfines has unbounded length it must be passing must be must be passing for the origin so composed by alfine for the origin only possibilities are guys like this for the origin but actually we are dealing locally dealing with a situation like this because also in this situation isolated curves there are no regions that can collapse so we are in the other situation only curves can collapse and they must be isolated otherwise multiplicity one conjecture is false so locally you have a collapse of an isolated curve so locally you are in this situation always so if you local you are in this situation and you take a blow up of things like this well you have very few possibilities actually you can you can have at most four alfines so at most you find something which is composed by four alfines moreover the fact well the fact that the the convergence is in c1 tells you that you must have 120 degrees 120 degrees here and here and again there is another lemma tells you that at the end again you can only have this situation 120 120 60 and 60 so the only possibility when you have a tree are the old ones so single line empty set single line triple infinite flat triode more this guy extra so four a tree without assumption on the curvature you only have an extra guy because of the topological structure that forces the shrinker to have this structure you only have an extra case to deal with in this in the argument that I used to yesterday so again you put yourself in a in a you take this guy is the this curve here is is vanishing you put yourself close and close to the point of vanishing what you see is that your guy is getting close to this so which means that even enlarging you are getting close to straight lines here and again you use a similar argument to the one we used for the trio in order to say that at some point you are too flat the other is not enough curvature in order that the curvature which is inside here goes to plus infinity fast enough and you use a very similar argument the one that I roughly presented yesterday that okay the curvature inside here can make well can go to plus infinity but it took too much time at the end you find out an estimated tells you that the coverage is going to plus infinity after some interval but this interval is larger than the interval given by big t the time of singularity so you conclude that the curvature at the big t is not going to plus infinity and again as yesterday as a heuristic principle yesterday I said when you find out the blow up which is flat there is no singularity the curvature is bound and also in this case we are adding to the the old list another possible blow up which is actually flat so it's not about to expect and actually it is like that the your curvature cannot go to plus infinity and in fact in this situation by this argument at the end you conclude that the curvature is bounded in the situation of a tree where some curve is collapsing down it is more or less it's very rough but otherwise I will spend a lot of time in showing the the technicalities of this but this is more or less the idea the more ideas that if you get the flat blow up curvature is bound without the consequence of this well a very first consequence is that what yesterday called type zero singularities actually exist singularities without with the bounded curvature that actually you can also imagine that possibly they cannot exist because if you consider a very long rectangle but the minimal connection between the four points here it's not so difficult to show that it's given by this guy 120 here 120 and a long line in the middle so now suppose change color that instead our initial network is done like this let's make it regular so angles are here 120 120 and run the evolution for this network as I said yesterday if the evolution doesn't develop any singularities you expect and that through that you converge asymptotically to the minimal connection between the fixed end points so if you don't have any singularity of this pink network you should get you should converge to this guy here but that's clearly impossible because they have a different apology in a way different structure they are not equivalent if instead so at some point a singularity must happen and when it happens since this guy is a graph by this theorem must be or this way the curvature must be bounded so there must be one curve collapsing and so and if you make it symmetric actually this curve must be a collapsing one because if you make it symmetric on one side and the other side it's not possible this curve collapses because otherwise also these other one collapse so the only collapsing must happen on this curve forming a four point like this with bounded curvature because of the theorem and there is starting by means of Ilman and Neverschultz's theorem produce something which will be something like this asymptotically you get the minimal connection but the point is that this must happen and this is a type zero singularity so you rigorously prove after this theorem that type zero singularity must really happens in some situations let me also say that all this can be localized which means that the theorem can be localized so a easy corollary is that if the network is locally a tree maybe over here there is some regions but locally a tree means that uniformly in time in that neighborhood the network is always a tree then you can use the theorem that says that whatever happens inside here inside this your network even if you find a singularity the curvature is still bound so where the network is a tree in this area even if it is not a tree globally the curvature is there in that neighborhood is bound and actually this is more than a mark if no region is collapsing actually the network is a tree because you can always shrink your neighborhood and without collapsing you only see a tree and as a consequence if no region is collapsing locally locally no region is collapsing locally the curvature is bounded and this is more this is actually the the converse of what we saw before that if a region is collapsing curvature cannot be bounded in this case if the region is not collapsing the curvature is bounded that tells you that bounded singularity with bounded curvature coincide exactly with this kind of vanish and curvature with a bounded curvature coincide with the collapsing of a region and now also the scroller can be clearly localized and now we deal with the collapse of a region the other family of singularities in this situation we assume so these two guys are equivalent after this argument so we have collapse of a region and unbounded curvature one implies the other so curvature is unbounded and even is the one multiplicity one is true again you perform a procedure what you find out then now you are no more it's so nice situation that you can only find lines infinite flat triads these guys composed by two lines crossing at 120 degrees but now you find a lot of things because every shrinker is a flow by mean curvature think of these guys suppose you have this guy is flowing by mean curvature is shrinking to a point and vanishing if you do a blow up at the shrinking point you find itself so you really find these guys for instance this one and this one the or the blackest blue bucket spoon or all these other possibly you can imagine that if you fix this guy at the free end points on it on a circle and let him evolve you simply look see him getting down to a triple to a triad to a flat flat irregular tree so there are a lot and classification in order to understand the singularity is important because of this because of this reason and clearly the more your original network is easier it's from a structural point of view the more you can find a small number of these possible network because as I said taking a limit to top the structure of the top of the complexity of your network simplifies because maybe there are taking a limit a blow up limit means that you are enlarging things so maybe some parts with higher complexity you are sending them to infinity so your complexity is getting is getting lower actually in the case then the region collapsing you have you see macroscopically your regions collapsing then you enlarge and what you actually what actually happens is that in a way the two things compensate and your region you after the rescaling the dynamical procedure of whiskey and your region is always there past the limit you find it in the limit and actually in fact what you find in the limit it's kind of it's the memory of the region which is collapsing down that you don't see without the rescaling procedure because if you look at think of things only without rescaling you see this region going and you if you're able to take a limit in the limit the decision has gone you have no idea how was the region if it has a five or four ages you only see a point where the region has disappeared instead the in the in the shrink in the block shrinker limit shrinker actually when you see this region it's what actually collapsed down and also if you find if for instance find something like like this supposedly this three for a shrink at these three lines are unbounded these three lines are it gives you the limit tangents of the non-vanishing curves which are the curves arriving at the region disappear suppose for instance we are something like this and now this region is disappearing what happens here that these three curves are going to be something here here the original hay collapsed inside this point this is time t and this is time at the limit time but you take take the proceed the block procedure on this guy what you see here is that you get something like this as a block shrinker and the direction associated these three alfines are exactly equal to the direction you find in the limit so the the the unbounded curves and mounted alfines here give the direction of the limit tangent of the curves which are not vanishing and arrive on the region which is actually collapsing down this is more or less what I said unfortunately in this case with unbounded curvature unbounded curve bounded curve is very good because it gives you immediately compactness in c1 here you have no control on the curvature so you don't have seen one compactness and moreover at the moment we are not able it's a contracture to show that when you take the limit of this network to to have something when t goes to big t you get something but actually we are not able to to show that it's a unique guy like instead I was able to to do before because before because of the argument about the Cauchy sequence bound on the velocity in the case of bounded curvature so this is we don't know and in fact at the moment we let it we conjecture it through that actually there is only a single guy that you can find out as a limit but it's a apparently hard open problem assuming such uniqueness actually what can we prove that possibly after reparametization there is a vanishing part of the evolving network in this case these three curves bound in this region that collapse to a point like this in this limit network s of t and a non-vanishing part that actually converge in c1 actually two curves that arrive that concourse actually at this multi-point now here you find the triple a triple junction only because the region here has as free edges but see other examples and also in this situation you are not guaranteed that this angle are 120 can be whatever another example suppose you have something like this so you have one two three four and what you get is something maybe something like this with four curves depending on the number of curves arriving from on the region which is collapsing down far from the triple junction as before because of local regularity or a car we can estimate for mean curvature flow what you get is that the the convergence of this curve if you are a little far from the triple junction here it's actually infinity and the curves are infinity but the best you can say about the curvature is that if you measure the distance of some point x here let me call it p here to the triple junction in arc length well the curvature here it's a small o of the distance of one over the distance so getting close to the triple junction to the to the multiple junction that you get in the limit the curvature can go to plus infinity but not so much smaller than one over the distance to the to the multiple junction this is a qualitative a quantitative very important estimate in this situation and this is an example of what can happen this is a nomothetic situation when you know very very special situation which is homothetic and why I was underlining this this estimate because actually the idea is that when you assuming the uniqueness conjecture when you find out something like this in the previous situation the guy in the situation with bounded curve the guy that we found as a limit time big d was with bounded curvature in this case the curvature is not bound and if you look at the at the theorem of in my neighbor's truth well actually I didn't write it clearly actually the the in the in the version which is up now published or at least on archive actually you ask that your curve have bounded curves apparently you are able to restart the flow in the situation of the of a tree but not in this situation where your curvature is unbounded then in a personal communication and without a written proof but I'm personally convinced and the argument that should be there where in this theorem you can have actually put as an initial network something under this only this condition so weakening a little bit the condition that are present in the paper on archive where they ask for bounded curvature or all the curve in order to have this bracket flow bracket flow that possibly up to now you saw the definition instead you can push the the proof in order to deal also with a case when the curvature has this kind of growth so actually it's an unpublished an unpublished proof of this of an extension of this theorem you can actually restart the flow after this after this limit s of big t under the uniqueness condition also for in the second situation of collapse of a region okay so let me get back and let me conclude maybe a little bit early with the a big picture of what we've done up to now that possibly resume everything that we've done kind of lien scheme okay like this okay on this here I put hypothesis on the singularity possible blow up network shrinkers and their conclusions okay first case that with yesterday the first case that we discuss it when when on the singularity we assume that all the length of the curve were bounded by some epsilon larger than zero uniformly and we assume it also in order to get a contradiction that the the supremum of the curvature well the curvature was bounded by some constant bounded a plus infinity what we work it out that the possible blow up must be either empty or actually at the point where there is a singularity embedded infinite length and regular hold only triple junction and 120 degrees only three points 120 degrees all of this under let me write in red multiplicity one condition under m1 multiplicity one condition it is one condition entered here embeddedness of the limit after you got this what are the possible shrinker with all these qualities all these properties actually a line for the origin only okay or empty set line for the origin or infinite flat only these two conclusion that we we got at the end that no singularity at all was a contradiction conclusion singularity by means of white theorem or the extension that we did with the manovaga and mani or the extension the analogous extension of ilman and nevus sheuls k in this situation is bounded so there is no singularity k stays bounded which actually was the idea to get this to get this proof you assume k was not bounded you get a contradiction at the end second situation let me write it like this inf of some curve is to zero so some curve the length some curve goes to zero and curvature bounded that actually I call type zero singularity singularity with bounded curve type zero single something that possibly I didn't say that in this situation of bounded curvature all the length of the curves of the network converged to something so not only the inf of some curve is zero but actually some curve the length of at least one curve must go to zero but really is going to vanish so there is at least one curve such that li was to zero what we discover we discover what are the possible blow up must be the empty set as usual embedded again no curvature because since we are doing a rescaling bounded curvature implies that the curvature in the limit is going to zero which also means segments because there is no curvature they are what we call the generate well our shrinkers and no regions because the argument that if you're bounded curvature no region can collapse so locally you are dealing with a tree without region collapse so in the limit you cannot have regions because the limit it's easier just less complexity than the approximating so no regions the generate means here that take a regular shrinker and possibly collapse some parts some straight parts actually in this situation because we are dealing with straight segments okay if now classify if you forget about every part forget about the part before if you classify regular shrinkers with no we're composed only by segments straight with no regions and with this idea that if they hide something which is that they are something that can vanish yeah you have the same as before this guy like we saw before so there is only one more in that situation and the conclusion is that actually you always have k bounded by an argument similar to these ones and you can describe the exactly what happens so it's standard transition this guy becomes two curves forming angles 120 and moreover also are isolated so it collapse the lapse of isolated curves with bounded curvature so this is s t and this is s big t final case again if of length of some length 50 and k infinity unbounded so which means that here we are dealing all together with type one and type two singularities I didn't separate the two cases actually we all have a conjecture and when I say whole let's say at least me, Matteo Novago, Alexander Pluda, Felice Schulze, Tommy Mann and that type two singularities for the motion of embedded networks are not there they don't they cannot be developed like the same for the motion of a single closed curve embedded type two singularities for curves in the train can only appear if you consider the motion of immersed curve possibly a curve like this very possibly when this loop will shrink will develop a type two singularities but this is only possible if your curve is actually immersed not embedded so if we look at this guy look at this situation type one and two because up to now it's a conjecture only conjecture type two singularities is the last case we discussed again non-empty or embedded curvature non-zero regions around again they generate so possibly there are some hidden parts and there can be a lot and actually as I said and you can find on the survey that I uploaded on the as a material for the first lecture there are several conjecture about many of them coming from Tommy Mann and about the what is what can be in this class actually the strongest conjecture that you can imagine is that this class is actually topologically finite so the possible structure are actually finite but anyway there are a lot and they are for sure they contain regions number a finite number of regions actually and again also here we have as usual M1 otherwise the conclusion is not there and again conclusions which is the last conclusion that I showed you under under the multiplicity the uniqueness assumption uniqueness of the limit again let me also underline that possible blow ups here at the moment non-unique again you can conjecture that the blow up is unique but even in the smooth case of mean curvature flow in several cases the problem of uniqueness of the blow up in the rescaling after rescaling procedure it's still open also in this case we conjecture that actually the possible blow ups could be unique but at the moment we are not able to prove it could be non-unique here instead we are asking if you remember uniqueness for the limit network when t goes to big t and the region is collapsing in this situation you can have a lot of things like a lot of things you can find here so stuff like this that actually globally are only c1 only c1 c infinity far from from far from the triple the multiple junction and with curvature or small o o1 over d also let me mention another conjecture here that actually we don't believe too much this estimate in the sense that we actually hope also in this situation k is bounded this is not absolutely necessary for restarting the flow like I will try to do tomorrow because actually the theorem of mean man and emissary should that could be extended to this situation so we can also in this situation this case is clearly in the hypothesis of their theorem this case apparently is not in the published version in the public version on archive but in an unpublished version you should be able to use the theorem also in this situation here if you have this estimate on the growth of the corvator getting close to the multiple junction okay just to conclude one minute I hope that apart with the analysis of the network for flow I hope I convince you of the power of the power of these techniques of using blow up in getting information on what happens but the blow up tells you about something microscopically close to the singularity but then what you really want to have information macroscopically on your flow and you see that at every step from information here you get information on the blow up then here you're trying to use this information to classify and hopefully restrict strongly restrict the possibility of the blow up that you find out hoping that you are good enough in using this information to classify what appears here and then once you classify you use the microscopic information that gave you knowing what you can find to get some conclusion on the macroscopic behavior of your flow this it's a very good scheme in general outside the network flow for mean curvature flow rich flow network flow or other kind of flows you can have in mind possibly in other situation we have different way to do blow up different way to do classification that usually this part it's usually where the geometry enters you use the geometry in order to classify the object that has these properties and write them fill the the holes here and hope that they are not too many when they are too many like here you see it's quite clear the last things you find here the more you are precise in the conclusion here this is more or less the philosophical message behind this scheme of work okay I think I can conclude here and see you tomorrow


Hello, thank you for joining me, Monica Wahee, in your journey as you learn to use SAS on Demand for Academics, otherwise known as SAS ODA. In this video, I'm going to show you how to look at the contents of a dataset using PROC contents in SAS ODA. Okay, here we are in SAS on Demand for Academics or SAS ODA environment. If you need help setting up your free account in SAS ODA, you can take my free online course called APLI, getting started with SAS ODA. I'll link you to it in the description. Okay, so assume you got started with SAS ODA and now you can open a program window. And let's say someone gives you a SAS dataset. Actually let's say I give you a SAS dataset and tell you to get it into SAS ODA. Actually, if you don't know how to get this dataset into SAS ODA, I have a video I'll link you to in the description. But go out to GitHub and get this dataset and put it in. So this is our dataset that I put in SAS ODA. It's called CHAP 6 underscore 1. See this folder X? I made that and uploaded the dataset to it. See this libname command? I use this to map to X. This is all in my other video. Okay, let's first look at this data step. We see it says data brfss underscore a as the first line. What does that mean? It means we are telling SAS to output a dataset named brfss underscore a into the work directory, which is SAS's working memory. That way, the processing will run faster, but the downside is that the working memory gets wiped when we log out, so we can't do anything enduring. The next line is set X dot CHAP 6 underscore 1. This refers to our dataset. So this tells SAS to read that dataset called CHAP 6 underscore 1 in the X libname into memory. So those data step commands just get this dataset into SAS's working memory, naming it brfss underscore a along the way. But what is in brfss underscore a? Let's see by using PROC contents. So this code runs PROC contents on brfss underscore a without any options. Let's run this data step code and this PROC contents code and see what happens. Let me highlight and run. Okay, so this sends us to the results tab. Let's look at the top output table. Here are two important pieces of information in the top of the PROC contents output, dataset name and number of observations. If you are processing lots of datasets over and over, like you do in a data warehouse, you are always running a PROC contents and you are always renaming datasets and changing the number of rows in them. That's why you should really look up here first before you read the rest of the output. You want to make sure you are reading the contents for the right dataset. Okay, let's scroll down to the last table, which lists our variables. Wow, look at these variables. These variable names are something else. See this number sign and this column titled variable? This says the number of the column in the dataset, like the first variable in the list, which is named ADDEPEV2, is actually the tenth column in the dataset. So how did SAS know to put it first? Well you might have noticed that default PROC contents lists the variables in alphabetical order. That's why ADDEPEV2 is first. Okay, let's use an option on PROC contents to make it list the variables in the order of the columns. Let's go back to our code. See this next code? This code has an option after I state the dataset. It says VARNUM. That's the option to make PROC contents sort in order of variable number. Let's highlight and run this and look at the order the variables come out in this time. Aha, this looks more normal now. Notice how the first variable is underscore state. This is a pretty important variable in the dataset because the dataset is a national health survey in the United States. So the state variable is very important. You can see the other variables in order 1 through 12. See these columns? Type, LEN, FORMAT and INFORMAT. In this output they don't look very helpful. They are helpful when you are trying to change the variable type as you can see in the type column or the variable length as you might suppose by the LEN column. Format and INFORMAT are also SAS functions that have to do with variable formatting. Again, these concepts are not so important in this video because we are not trying to manipulate the format, type, or length of these variables. But if we were, we would really need to be running PROC contents a lot, like each time we did an operation to see if we were adjusting the type, length, format, and INFORMAT correctly. Hey, if SAS looks hard and you want to be easy on yourself, take my free online course on how to use SAS on demand for academics. All these lessons are based on my popular book, Mastering SAS Programming for Data Warehousing. I'll link to these in the description. Thanks for watching and have a wonderful day.


Lecture 24, As-Salaamu Alaikum. Welcome to the virtual university's course on business and technical communication. In the last lecture, we looked at planning long reports and today we will look at writing long reports. In this lecture, we will describe how organizations produce formal reports and proposals. You will learn to prepare all necessary parts of a formal report. Also, you will learn how to select and prepare the visual aids to include in a formal report. We will also look at assembling all the parts of a formal report in the proper order and using the appropriate format. Let's have a look at report production. Planning formal reports and proposals conducting the necessary research, organizing the ideas, developing visual aids and drafting the text are all demanding and time consuming tasks and all of these go towards the production of a report. After careful editing and rewriting, you still need to produce a polished version and all these things will tie together to produce that final polished version of your report. How to produce your report also depends upon the equipment that you have available to you. Personal reports automatically handle many of the mechanical aspects of a report preparation. Be sure to schedule enough time to turn out a document that looks professional. If you keep all your time to writing the report and planning the report and writing the report and you don't keep give yourself enough time to edit or rewrite the report, then at the end when you have your deadline close by, you will be panicking and the final look of your report will not be professional enough. When you're composing a formal report, you need to keep in mind that a professional report conveys the impression that the subject is important. The three basic divisions of a formal report are the prefatory parts, the text and the supplementary parts. Let's have a look at each of these separately and see what they mean and what they include. The prefatory parts may be written after the text has been completed. Many of these parts include things like table of contents, list of illustration, synopsis, etc. And these are easier to prepare after the text has been written. Other parts can be completed at any time. Some of the parts of the prefatory text include the cover. Many companies have standard covers for reports made of heavy paper and imprinted with the company's name and logo. You need to put the title on the cover and the title should be informative but not too long. You don't want to intimidate your audience with a title that is too long, awkward and unwieldy, difficult to understand as well. Other things could be the title fly and title page. Title fly is a plain sheet with only the title of the report on it. The title report includes four blocks of information, the title of the report, the name, title and address of the person that authorised the report, the name, title and address of the person that prepared the report and the date on which the report was submitted. You also have the letter of authorisation and letter of acceptance. If you received a written letter of authorisation to prepare the report or proposal then you may want to include that in your report. Also remember that a letter of authorisation usually follows the direct request plan. So we looked at requests earlier and if there is a letter of authorisation it will follow a direct request plan where the person who has asked you to write the report will be following that plan and you will include that letter with your report. If you are writing a letter of acceptance then you will use a good news format, a good news letter format to accept that letter, to accept that authorisation and to say that yes you will indeed prepare that report and that also you may attach with the report if you feel necessary. Then there is a letter of transmittal which may be used if you are sending your report somewhere. This letter of transmittal conveys your report to your audience. You will use a less formal style for the letter of transmittal than the report itself. Generally, the letter of transmittal appears right before the table of contents. The letter of transmittal follows the same routine and good news plan as described in earlier lectures when we talked about good news plans. You will definitely have, if it is a long report, a table of contents which indicates an outline in the form of the coverage, the sequence and relative importance of the information in the report. Depending on where you have included specific information in the report and how that is listed in your table of contents that will highlight the importance of that particular piece of information as well. The table of contents is one piece, one prefatory part that is prepared after other parts of the report have been written or typed so that the beginning page number of each heading can be shown. Obviously, if you will write the table of contents first and then write the text later then you will not be able to put the page numbers in the table of contents. You will put the list of illustrations on a separate page if it doesn't fit on the same page as the list of table of contents. If you are using a list of figures and a list of tables separately and if they can't fit on one page then you can write them on separate pages. If you are dividing your illustrations in figures and tables then you will have two different lists and if you don't have two different lists on one page then you can write them on different pages but try to fit the list of tables and the list of figures on the same page. You will also have a synopsis or executive summary. A synopsis is a quick way to overview the report's important points and this is designed to give readers a quick preview of the contents. Because it's a concise representation of the whole it may be distributed separately to a wide audience. The synopsis can be attached to the report and it can also be given out to people independently without the report to generate interest in the report and then if they show interest and if there is a need to send the whole report to them then the report is sent out. The phrasing of the synopsis can be either informative or descriptive depending on whether the report has a direct or indirect order. An informative synopsis summarizes the main ideas whereas a descriptive synopsis describes what the report is about. For example an informative synopsis could be something like sales of super premium ice cream make up 11% of the total ice cream market and the descriptive synopsis for the same report could be this report contains information about super premium ice cream and its share of the market. You will use a descriptive synopsis for a skeptical audience and you will use an informative synopsis for other situations. If you feel that your audience is skeptical about what you are trying to say, then you will use a descriptive synopsis and you will not tell your findings in the beginning. But if you feel that your findings are easily accepted then you will use an informative synopsis and you will tell the information in the beginning. A synopsis or objective summary is a fully developed mini version of the report itself. Whatever is in your report it is intended for readers who lack time or motivation to read the complete report. A lot of the times you will be writing for people who are very busy and they will not have the time to read the whole report. So then they will read the executive summary and then if they feel that it is really important for them to read on then they will read the whole report or sometimes it might not be lack of time it might just be lack of motivation and even then your executive summary is then designed to help motivate the people to go on to read the whole report. So you will put enough information in a synopsis or executive summary so that an executive can make do without reading the complete text as well if they do not have the time or if you feel that they will not go on and read the whole report. Now coming to the text of the report you will need to make the decisions about the design and layout of the report and keep in mind that aids to understanding the text of the report include things like headings, visual aids and preview and summary statements. Headings are the most powerful format tool that are available to you. By skimming along from heading to heading the reader should be able to pick up the structure or outline of the report. And this process is easier if the headings are phrased and typed in a consistent way. Visual aids are also useful tools for calling attention to key points and helping readers grasp the flow of ideas. I catching graphics dramatize the high points of the message and informative captions explain their meaning. So make sure that you have graphics which are eye-catching which also are relevant obviously and then the captions that go with them are informative. It is also useful to preview summary points at the beginning of each major section or chapter and to sum them up at the end. Basically at the beginning of each section you need to tell the readers what you're going to tell them then in the middle sections you will tell them give them that information and at the end of that section you will tell the readers what you have told them. So in the beginning you tell them what will happen in the main text, a summary then you give them the main text and in the end you sum it up and recap the information in the entire section or chapter. The first part of your text is the introduction. The introduction to a report serves a number of important functions. It puts the report in a broader context by tying it to a problem or an assignment. It tells the reader the report's purpose. It previews the report's contents and organization and it establishes the tone of the report and the writer's relationship with the audience. The introduction deals with authorization. It talks of when, how and by whom was the report authorized, who wrote it and when it was submitted. All these aspects are covered in your introduction. It talks of the problem and the purpose which means the reason for the report's existence, what is to be accomplished as a result of the report being written, what is the purpose, what is the problem, why has the report been written basically. It also talks about the scope of the research that has been undertaken, what is and what isn't going to be covered in the report. The scope indicates the report size and complexity. What are the things that have been included in the report, what have been not done, what is the size of the report, what are the complex issues that have been covered. This is all its scope. And also the background, the historical events and conditions that have led up to the report. If a previous type of report has been written, and this report is being written as a result of the report, or some work has been done because of which it has been necessary to research and write the report, it will be mentioned in the background. You will also indicate the sources and methods that you have used. The secondary sources of information used and the primary sources such as interviews, surveys, experiments, observations. This will be highlighted as to what type of sources have been used in the report. And there will be a brief introduction of the definitions that have been used. A brief introductory statement leading into a column of terms used in the report and their definitions. It's important that if there are going to be any technical terms used then they need to be defined. So in your introduction you can have a little column which gives the definitions and obviously that column will be introduced by a sentence which says that these are the terms used and this is what they mean. You will also talk about the limitations of your research in your introduction. Factors indicating the quality of the report such as if the budget is too small to do all the work that should have been done, if there were time constraints and other events beyond your control that led to any limitations in the report, anything that was lacking in the report, then you will mention that. This is important because if you do not, then it seems that you were careless. But if you do mention it, whatever limitations you had, then you are letting the audience know that you are aware of these things, you have not overlooked them, but you have not been able to deal with them because of these factors. If you know that there are things that should have been covered in your report, it could not have been because the money was low, there was no time, there was no cooperation or whatever the factors are, then you should mention it yourself instead of being able to pick out the readers that these are the things that are missing in this report because it will feel that you have not completed your work. Also the report organization is referred to in the main introduction. The organization of the report along with the rationale for following that plan, how you have organized the report and why you have done it, what was the rationale for following that plan? You will also tell this in your introduction. After this you will come to the main body of the report. It consists of major sections or chapters that analyze, present and interpret the material that has been gathered as a result of your investigation. Whatever material you have gathered, you will present it in chapters, sections and in those chapters and sections, that material will be analyzed, presented and interpreted. One of the decisions to make when writing the body of your report is to decide how much detail you need to include. One of the major interesting things about writing reports is that generally when people are asked to write a long report and they are given a word limit in which to write that report, the first reaction they have is that this is too much and that this is too many words or too many pages and we will never be able to write so much. But then when they start actually researching and they get all the information material, then one of the major problems is that they don't know how to actually fit in all the information in the scope of the report. So then it turns the other way around. So you really need to decide what to include and what not to include because every report will have a limit and you do not want to include irrelevant information or information that will lead to the reader losing interest in your report. Another decision to make is whether you want to put the conclusion in the body or as a separate section or the third option could be that you could put the conclusion in both, you could have it in the main body and then have a separate concluding section as well. Final section of your report tells the reader what you have told them, basically the summary, conclusions and recommendations. In the summary you will state the key findings of your report, these will be paraphrased from the main body and stated or listed in the key order in which they appear in the body. In the conclusions, the writer will be presenting his or her analysis on the findings and what they mean. These are the answers to the questions that lead to the report in the first place. The third thing would be the recommendations. These are opinions of the writer based on reason and logic about the course of action that should be taken. If the report is organized in the direct order, the summary, conclusions and recommendations are presented before the body and are reviewed only briefly at the end. In action oriented reports, you will put all the recommendations in a separate section and spell out precisely what they mean. Now you will also have notes in your report. When you are writing the text of your report, you decide to acknowledge your sources. You will give credit where credit is due. Because if you don't, then it becomes plagiarism, then it becomes stealing people's ideas and putting them down as your own. If you have got any other information, then it is important to tell you where you got that information. Plagiarism basically occurs when one person misappropriates without permission any ideas or facts or words that were originated by others. So if there are any ideas, facts or words that were actually said by somebody else and you are using them without their permission and without referring to them or without giving them credit, then you are actually stealing what they have said. In general, you have flirted with plagiarism when your business documents fail to alert your audience that you have either repeated someone else's information word for word or paraphrased another's material too closely or lifted a series of phrases and put them together with your own words or borrowed a unique term that originated elsewhere. So then you are plagiarism. Basically, if you are using some other words word to word or paraphrasing someone else's material, but it is exactly the same meaning and you have just changed a few words from here to there. Or you have taken some phrases from here to there to a person's work, to a person's book, to a person's report, to different phrases, you have joined them and you have given them the impression that this is what you have written or there is a term or a concept that someone else originated and you are taking it in this way as it is yours. All these things come into plagiarism, cheating, stealing. Although your company may have more specific guidelines about plagiarism, how to avoid plagiarism, I am going to give you a few general tips on how to handle situations that commonly arise. If you are repeating information from another company document, that is something that can happen and that is plagiarism. If you reuse information appearing in another report, your audience may believe that you have independently verified material and eliminated any errors in the original report. So, if you have any information that has been written in a previous report and you are using it in your own report again, this audience will get the impression that maybe you have researched it. Otherwise, they will feel that it was in the first report and you have verified it and given them more information about it. That is why you are using it in your report. Or if you are using the same sources as another document, consulting somebody else's sources for further information is perfectly acceptable, but plagiarism deals with the way the information is reported, not whether you access the sources used by someone else. You should be aware of the ways that your colleague or someone else has used it. Not that the way they have used it, you should use it the same way, the exact ideas, the exact conclusions you should draw, then there will be no difference between what has been done in your work and what has been done before. It will only change the meaning of the words. So, this comes into plagiarism. Because you have taken all the ideas for them and you have written them in your words. Also, another commonly arising situation is when people repeat information that is protected by copyright. If you repeat information from books, published articles and other copyright materials, be sure to avoid infringing on the originators' legal rights. So, be careful to talk to your company's management before you repeat information that is protected by copyright. One approach, especially for internal reports, is simply to mention a source in the text. For example, according to Dr. Raftaab of Kangaram Hospital, hip replacement operations account for 7% of all surgery performed on women aged 65 and above. Now, this is perfectly acceptable. It's an internal report within the hospital, for example, or within an organization and then you can refer to a specific person who has found, who has done a research and found some results. But if your report is to be distributed to outsiders, you should include extra information on where you obtained the data. Then you will need to say, according to Dr. Raftaab of Kangaram Hospital, as stated in a conference on hip replacement, for example, or in the details of the conference, you will also need to give it to you. When you're using visual aids in your main body in the text, you face the problem of choosing the form. What form of visual aids to use, which best suits your message? Visual aids are a lot of things. Figures, graphs, tables, pictures. But which visual aids do you have to use, which are better highlighted depending on what you're writing about? Also, good business ethics, demand that you choose a form of visual aids that will not mislead your audience. Visual aids could be tables. When you have to present detailed specific information, you will choose a table, which is a systematic arrangement of data in rows and columns. You will use tables to help your audience understand detailed information. Although formal tables set apart from the text, information which is necessary for conveying complex information or data which is necessary for conveying complex information, some data can be presented more simply within the text. Data is such that it cannot be explained within the text, then you can explain it better in a formal table. Tabular information can be introduced within a text without a formal title. If it's information that you're using within the text, you don't really need to give a formal title. But if you're giving information in a more formal table form, then you'll have to give it a title as well. If you're only including one or two lines of information in a table text, then you don't need to give a formal title. The preceding or later text will explain it. Other visual aids could be line and surface charts. A line chart illustrates the trend over time and plots the interaction of two variables. You will basically use line charts to indicate change over time. A surface chart is a kind of a line chart with a cumulative effect. All the lines add up to the top line, which represents the total. And this form of chart is useful when you want to illustrate the changes in the composition of something over time. How something was composed? Another type of chart, a chart which is commonly used, is a bar chart. A bar chart is one in which amounts are visually portrayed by the height or length of rectangular bars. And bar charts are valuable when you want to compare the size of several items at once. You want to show changes in one item over time, over different years, or you want to indicate composition of several items over time, or show relevant sizes of a whole. of a whole. In a pie chart, numbers are represented as slices of a complete circle. Although they are less versatile, pie charts are nevertheless valuable as visual aids as well. When composing a pie chart, try to limit the number of slices to no more than 7. Other forms of visual aids are organization charts and flow charts. You will use organization charts to depict the interrelationship among the parts of an organization. You will use a flow chart to show a series of steps from beginning to end and also to show relationships. Once you have finished writing the text of your report, you will conclude it with a summary. And if appropriate, you will include a conclusion and recommendations. In a summary, you will recap the findings and explanations that have already been presented. You will place conclusions and recommendations in their order of logic or importance, preferably in list format. If you have any conclusions or recommendations, you will place them in the same order of logic. If you have any conclusions that you want to tell them later, you will place them later. The most important conclusions and main findings that you are looking for, you will tell them first. If you have any logical options, you will place them in the same order of logic. If you think that you will place them in the same order of logic, you will place them in the same order of logic. Preferably in list format, you will place them in the same order of logic. Also to induce action, to make sure that the readers do what you expect them to do and what you want them to do, you should explain in the recommendations section who should do what, where and how. If appropriate, also point out the benefits of the action to leave the readers with the motivation to follow on the recommendations. You will do it because they are beneficial for your company or maybe the people who follow the recommendations are beneficial for them. So clearly point out the benefits of following these recommendations so that the readers have the motivation to follow your recommended action. Now as we come to the end of this lecture on writing formal reports, I am going to show you some slides that will give you examples of some of the parts of a long report. Most of these are the prefatory parts that we've talked about earlier in this lecture. If you have any questions, you can email us and we will be happy to guide you according to how and where these parts can be used. Let's have a look first at the title page of a formal report. As you can see it has the main title which is not very long and it's captioned in a way that it's easy to read and easy to understand. Then it says who it was commissioned by and who it has been written by and the date. Alongside that is the letter of transmittal, the letter that goes along with the report when it is being sent somewhere and that is like a normal routine good news letter. The table of contents of formal reports will follow a pattern where you will write about the main sections, the main prefatory parts that come with the report, you will list the initial parts, the abstract, the table of contents, etc. And then you will talk about the main body and as you can see there are headings and there are subheadings and the page numbers are given alongside them. If there is a list of tables and figures that is to be included then that will list what page each figure appears on and what that figure is about. For example, if it is a figure showing the technical marketing process or whether it is the employment outlook for technical marketing, whatever it is that the figure is about or the table is about, you will give the name. It's not sufficient just to say table one figure, one figure, two figure, three, etc. Also make sure that whatever headings in the contents page or whatever captions for the tables and figures you are using in the list, they tally with what is within the main body. It is not that you have given a title in the list and actually you have given a title on the figure one. The initial paragraph talks about the main analysis in the report, the main findings and then it ends with a list of recommendations. Now this abstract is enough for anybody reading the report to know what the writer found and what needs to be done as a result of those findings. There can also be a partial glossary. We talked about defining terms earlier and this is what the glossary would look like. You take the technical terms and you define them. You can also have appendices to your report. Appendices are information material that is put separately attached separately to your report and it is not part of the main text or the main body of your report. In such a case if there are more than one or two appendices then you will also have a list of appendices. When you give your list of tables and figures then you will also have a list of appendices if they are worthy of being put in a list. Whatever appendices you have you will refer to them within your text so that they can be some relevance to having them in the text. Appendices will come at the very end of your report but they need to have a reference in your text. And also generally appendices are titled by appendix A, appendix B, etc. And if need be then there is a caption as well. If there are tables or articles whatever informational material that you feel is necessary but not so necessary that it has to come in the text. A lot of the times there is material that you writers feel the reader would benefit by reading but they don't want to include it in the main text because it breaks the flow. So then they just refer to the main finding of the material but they keep the main material as an appendix. In this lecture you learnt to basically look at the organization of formal reports, all the parts that are necessary for a formal report. And you also looked at the selecting and preparing the visual aids to support information that is given in a report. And we also talked about what type of visual aid to use for what particular type of information. For what type of information and for what type of visual aid we will use. We also looked at assembling all the different parts of a formal report in the proper order. And what type of format to use. We looked at the prefatory parts, the text, the concluding parts, how they would function together, how there would be references from one part to another if need be and what parts serve what purpose. If you have any questions please feel free to email us. The address is englishatvu.edu.pk. Until next time, Allah Hafiz.


My name is Lee Byer. I'm the head of the experiment station here at Uri and this afternoon we'll be cooking congee. Now congee is a very popular recipe through and much of South East Asia and East Asia. It's a nourishing and savoury start to your day. Some would say it's something like Asian comfort food, Asian chicken soup. The recipe we'll be preparing today was shared with us from one of our good friends in Laos, Deng, who's the owner of Kung's Cafe Laos in the capital city of Laos, Vien Chow. There's three components to congee. The first is the stock. Now we've pre-prepared some stock, as you can see. Now the components of the stock are a whole chicken, a whole onion, coriander and ginger. And this has been simmering for 30 or 40 minutes now so it's almost ready to be used. The raw ingredients we use again just a whole chicken, ginger, whole onion and coriander. Some would call it cilantro. For those that haven't got so much time you can cheat and get powdered chicken stock or chicken cubes. Or you can choose a pork stock or a fish stock or a vegetable stock. Choose the stock of your choosing. So I must admit on most occasions we cook congee, we usually use the cheats way. The second component is the rice. So according to the recipe of Deng, the owner of the cafe, she likes to use 20% glutinous rice. Which gives it a more creamy texture. So we've got roughly 20% glutinous rice and 80% non-glutinous rice. I have seen recipes where they've used 100% glutinous rice. So I think the ratio of rice is somewhat to your choosing. And finally congee is simply rice soup. It's got the texture of soup. So once you've combined your stock and cooked your rice in the stock, it's then awaiting the condiments of your choice. So we've got some basic condiments with us here. We've got more coriander, more cilantro, we've got spring onion, we've got fried garlic. We've got some fish sauce to your taste. Or just some classic seasoning sauce. And likewise you can choose some cracked pepper. And finally, particularly if you're a vegetarian, you might choose to top it off with one or two boiled eggs. So there's your basic congee recipe. The choice of condiments is limitless. There are many, many other choices loaded really in by your imagination. So we've prepared our stock already. So now we will move to combine the stock with the rice. And then it will have to cook for 30 or 40 minutes until it's ready to serve. Water is roughly 6 to 1, which is a lot. So the ratio of rice to stock is roughly 6 to 1. So you've got to put in quite a lot of stock. Much more liquid than you would usually use when cooking standard boiled rice. So that's 4 to 6. So we'll start with a ratio of 6 to 1. I've changed my mind already. I think it's probably now 8 of these to start with. And we'll see how that works. So we'll switch the stove top on. Now we tip in some of the non-glutenous rice. Just a little left over. Quite that much. And we'll tip in some of the glutinous rice. Ideally it should have been pre-washed, but we won't do that today. Now we simply wait for that to come to the boil. Particularly if I use the right burner. And that needs to simmer for at least half an hour. And the last thing we need to do is give it a stir. Just to get the rice off the bottom of the pan. And then we'll suspend it in the stock. You can see already we probably need a bit more to get to our desired ratio. Don't be scared to put in a lot of stock. You would be amazed how much water the rice absorbs. Looks like a better start. Come back to it in about 30 minutes. So here we are about 40 minutes after we began the preparation. And our congee is ready. It's got the right creamy soup-like consistency. So we'll now tip it out into a bowl. In the last 10 minutes or so if you want you can add some finely chopped vegetables like carrots or bok choy or cabbage or something like that. Into the mix if you wish a little bit more variety. You can even crack an egg and stir an egg through the congee if you wish. One big bowl of congee today. We will need many spoons. Okay, now we bring it over to our condiments. And we might choose to put a bit of shredded chicken in. Open the reasonably fine pieces. Perhaps some boiled egg. Some spring onion. The green always makes things look tastier. Some more coriander or cilantro. My personal favourite. Some fried garlic. Some sauce. Some fish sauce as well.  And here we have the finished product. Congee. In Laos they call it kaupia kao which literally means wet rice. A perfect start to your day.


Welcome to the screencast provocatively titled, How do we know if a statement is true? So in the last video we defined mathematical statements to be declarative sentences that have a definite truth value.  But what about statements whose truth values is not so easy to determine? How do we determine truth? Well let's look at a case study here, and we're going to define an important kind of number here first. We're going to say that a positive whole number p is a prime number. If p is bigger than or equal to two, and if it only has two divisors, namely one and itself. So the numbers two, three, five, seven, eleven, one hundred and thirty-nine. My favorite prime number, eight, six, seven, five, three, oh nine, which was immortalized in a catchy tune from the eighties, are all prime numbers because they are whole numbers, they are all bigger than one, bigger than or equal to two, that is, and can only be evenly divided by one and themselves. And non-prime numbers include numbers like four and twelve, which have divisors more than just one and themselves. For example, twelve could be divided by one, two, three, four, and six and twelve. So here's a statement to think about, whenever p is a prime number, the number two to the p power minus one is also a prime number. Now this is a statement because it's a declarative sentence, it's declaring that something is true and the statement's truth value doesn't depend on opinion or on the particular value of p. It's saying that as long as p is a prime number, whenever p is a prime number, so is the number two to the p minus one. So this is a statement, it's a claim. Being a statement, it's either true or false and the question is, which one? Well, it's very hard to tell just by looking. Finding out the truth value of this statement is a mathematical problem and a solution to that problem will consist of two things. First, we have to clearly state whether the statement is true or false. That's gotta be part of the solution. And then, very importantly, we have to explain why the statement is true or false. And to understand the solution of this problem, as with the solution to any problem, we need to first experiment with the problem and play with it. So let's pick some prime numbers and just compute two to the p minus one and see what happens. And we'll keep our results in a table here. So when p is equal to two, that's the smallest prime number, then two to the p minus one is two to the two minus one. And that's four minus one, and that's three, and that's a prime number. When p is equal to three, then two to the p minus one is two cubed minus one. That's eight minus one, which is seven, and that's another prime number. When p equals four, oops, I can't choose that. I can't choose p equals four, now why not? Well, four isn't a prime number because it can be divided by two. And so it doesn't fit the description of the statement. The statement is claiming something is true if p is a prime number. It says nothing about non-primes. It makes no guarantees as to what happens if p is not prime. So there's no point in playing with this problem to choose p to be anything other than a prime number. So let's move on to the next prime, which is p equals five. Now two to the fifth minus one, two to the fifth is 32. 32 minus one is 31, and it so happens that's a prime number two. An x prime number up the list is p equals seven. Two to the seventh minus one is, let's see, two to the seventh is 128. 128 minus one is 127, and we can check and see that that's prime. So this is fantastic. All of our examples are working, so what does this mean? Let's take a quick concept check here to see what you think it means. This is a very simple concept check, but I have to think about the answer here. So the fact that the two to the p statement was true, for p equals two, three, five, and seven means what? That the statement is definitely true, or that the statement might be true, but we don't know yet. Now think about that question and pause the video and unpause it when you think you got the answer. So the correct answer here is b, the statement might be true, but we can't say that the statement is true for sure yet. Why not? Well, recall what the statement actually says. It says whenever p is a prime number, two to the p minus one is also prime. That word whenever is all important. It is saying that every example of a prime number that we or anybody else could ever produce should work. Four examples, which is what we've done here, actually five examples, p equals two, three, five, and seven, I guess it is four examples. That's just not enough because what about all the examples that we haven't tried? And in fact, speaking of those examples we haven't tried, let's go to the next prime number. The next prime number after seven is 11. And we can calculate that two to the 11th minus one, two to the 11th is 2048, minus one is 2047, and unfortunately 2047 is not prime. It's equal to 23 times 89. So it actually turns out that this statement that I made here is false because it claimed that every time p is prime, two to the p minus one is also prime, and that's simply not the case. The statement is sometimes true, but because it is not always true, we're gonna say that it is false. So it seems like we failed somehow, but we've actually been successful in solving our problem. We've determined that the statement that whenever p is prime, two to the p minus one is prime is actually false because it made a general statement about prime numbers. But we found a specific instance where that claim didn't work, a prime number p such that two to the p minus one is not prime. So one example was enough to prove the statement is false, but notice that no amount of successful examples would ever prove this statement true because even if we had a million examples that worked, what about that million and first one? So there's an important factor to take away about statements like this. We cannot prove a general statement to be true using a list of examples. Now those examples are useful because they help us to make good guesses as to whether a statement is true or not. They help us to believe whether a statement is true. But it doesn't constitute a proof. So let's recap what we've seen. Given the statement that has a definite truth value, but you don't know what it is, the main way to get a feel for its truth value is to play with the problem rather than try to solve it immediately. This has two positive effects. First, you'll understand the problem better. Second, you may stumble across something that actually answers your questions, as we stumbled across the fact that two to the 11th minus one isn't a prime number in the first problem. In either case, you'll be on a more sure footing to proceed further into the problem whether you believe that the statement you're trying to work with is true or not. We've seen that examples help us give a sense of the truth value of a statement. But examples don't really prove anything unless the example shoots down a general statement. That's all for now. Thanks for watching.