Welcome to today's lecture on measure and integration. This is the second lecture and in this lecture, we are going to cover the following topics. Algebra generated by a class, then we will look at algebra generated by a semi-algebra. We will also look at what is called the sigma algebra of subsets of x and sigma algebra generated by a class of subsets of a set x. Let us just recall what we called as the semi-algebra of subsets of a set x. It was the collection, so let me just recall, a semi-algebra c is a semi-algebra of subsets of a set x. If at the following properties, empty set and the whole space belong to it. And secondly, for a and b belonging to c implies that the intersection of these two sets is also an element of c. That is the class c is closed under intersections. And third property was that if a belongs to c, then this implies that a complement can be represented as a finite disjoint union of elements of the class c. That is, a complement is a union of elements c i, where each c i belongs to c and c i intersection c j is empty for i not equal to j. Such a class was called a semi-algebra of subsets of x. Then we looked at what we called as the algebra of subsets of a set x. So, a class f of subsets of a set x is called an algebra. If it had the following properties, namely empty set and the whole space belong to f, as in the case of semi-algebra. Secondly, a and b belonging to f imply that a intersection b also belongs to f, as was the case for the semi-algebra. And third property, which is different from the semi-algebra, which is a bit stronger than the semi-algebra is, whenever a set a belongs to f, this should imply that a complement also belongs to f. So, an algebra of subsets of a set x is a collection of subsets of x, which includes the empty set and the whole space. It is closed under intersections and it is also closed under complements. And last time, we looked at some examples of algebras and semi-algebras. So, today what we are going to look at is, the question is given a collection c of subsets of a set x, does there exist an algebra, which includes c. So, the given collection c may not be an algebra of subsets of x, it is an arbitrary collection. And we would like to know, can we find a collection of subsets of x, which includes this collection and is an algebra. Of course, there is one obvious answer, namely the power set. For example, the power set of x is always an algebra, because it is collection of all subsets of x. So, it has all the properties, namely it is closed under intersection and complements and includes the empty set in the whole space. And obviously, c is a subset of px. So, in some sense px is the largest algebra of subsets of x, which includes any collection c. So, we should modify our question, namely given a collection c of subsets of a set x, does there exist an algebra, which includes c and is the smallest. So, to answer that question, let us look at the following. So, let us look at the collection all f such that f is a subset of px, it is a collection of subsets of x and such that f is an algebra and c is contained in f. c is the collection, which is given to the universe, which may not be an algebra. So, let us collect all those collections of subsets of fx.  So, it has two properties. One, c is a subset of f, this collection c is inside the collection f and f is an algebra. So, first of all let us observe that px is an element of this collection. So, let us call this collection as say i. So, this collection of all algebras, which includes c is a non-empty collection because the power set of x, the collection of all subsets of x is an algebra and includes c. So, this is non-empty. So, implies that this collection is not empty. Now, what we do? Let us define a to be equal to intersection of all this f such that f belongs to i. So, let us take all the collections in this collection, all the algebras, which are members of this collection i and take the intersections. So, keep in mind each f is a collection of subsets and it is an algebra and we are taking the intersection of these collections. So, the claim is that 1, c is a subset of this collection a, which is obvious because the collection i of all the algebras f has that property c is a subset of each member. So, this property is obvious. Secondly, we claim that a is an algebra. So, to prove that a is an algebra, let us observe what we have to do. So, what is a? a is the intersection of all algebras f, which are inside i. So, first we want to show empty set belongs to a. Why? Because we know empty set belongs to each f because each f is an algebra. Similarly, x belongs to a because of the same reason because x belongs to f and f is an algebra. Secondly, let us take a set e belonging to a. Then that will imply by the very definition that e belongs to f for every f inside the collection i. But that implies because e belongs to f and f is an algebra that implies that e complement also belongs to f for every f belonging to i. But then that implies it is equivalent to saying this implies that e complement belongs to the intersection of all these f in i and that is precisely our a. So, we have shown if e belongs to a, then e complement also belongs to a. So, the class a is closed under complements and let us finally show that it is closed under intersections also. So, let us take two sets e and f belonging to a. Then that implies that e and f both belong to each f, f belonging to i.  So, that is crucial. So, that is being used again and again. e and f belong to the collection f which is an algebra. So, the intersection also belongs and hence e intersection f belongs to a. So, what we are showing is this collection a which is intersection of all the algebra which includes c is an algebra. So, we have shown the second property namely that this collection a of intersection of all the algebras which includes c is an algebra of subsets of the set x. Also, c is inside a and by the very nature because it is the intersection of all the algebras which include a, it should be obvious. So, obvious property namely a is the smallest algebra of subsets of x such that c is inside a. What we mean by that? That means that if, so this is same as saying if f is any algebra and c is inside f that implies that this f have to include a. So, what we have shown is that given any collection of subsets of a set x, if we define a by this a is the intersection of all the algebras including i then this such an algebra then this collection exists because there is at least one algebra which includes c namely the power set and it is a smallest. So, what we have shown is given any collection of subsets of a set x there is an algebra of subsets of x which is smallest and includes a. So, we have answered this question that given a collection c of subsets of a set x does there exist an algebra which includes c and is smallest. Yes, the answer is yes. So, let us state this as a theorem namely let x be any set and let c be any class of subsets of a set x. Let f of c denote the intersection of all the algebras a which include the collection c. So, f of c is the intersection of all the algebras that include the collection c then what we have just now shown is that c is a subset of x that means f of c includes c and f of c is an algebra of subsets of x. So, f of c is an algebra which includes x and it has additional property it is a smallest with that property that is if a is any other algebra of subsets of x is that a includes c then a must include f of c. So, this is not s of c it is f of c. So, f of c is called the smallest sigma algebra of subsets of x containing c and is called the algebra generated by the class c. So, what we have shown is that given any collection of subsets of a set x we can always find an algebra of subsets of x which is smallest and includes it. Let us look at some examples. So, let us look at the collection x is any non-empty set. Let us look at the collection of all singleton subsets of this collection x. So, c is the collection of all singleton sets where x belongs to x. We want to know what is the algebra generated by it and the claim is that the algebra generated by this c is nothing but all those sets e in x say that either e or e complement is finite. So, let us prove this how do you prove such kind of a assertion. So, we have got x any set and c is the collection of all singleton sets belonging to which are subsets of x and then we are looking at we claim. So, this is our claim that f of c is nothing but all sets a contained in x say that a or a complement is finite. .. So, let us observe. Does empty set belong to f of c? Yes, because by definition empty set is a finite set. Does x belong to f of c? So, for x to be an element of f of c either x should be finite which we do not know because it is an any set. It may or may not be finite, but we know it is complement which is empty set is finite and in f of c. So, by the second criteria a or a complement. So, x may or may not be finite but it is complement is empty set which is finite. So, this property is true. Let us check the second property namely if e belongs to f of c does this imply e complement belong to f of c? Is this true? Well for e complement to belong to f of c either e should be finite or e complement complement should be finite. So, this will be true if and only if e complement or e complement complement finite which is same as e complement or e complement complement is e finite which is same as saying. So, this is if and only if this is if and only if this is if and only if e complement belongs to f of c. So, e belonging to f of c is true if and only if e complement because our definition of f of c is symmetric with respect to a and a complement. So, the collection f of c of all those sets for which a or a complement is finite includes empty set the whole space it is closed under complements. Let us show the third property namely that if e and f belong to f of c then that implies the intersection also belongs to f of c or equivalently this is equivalent to saying that e union f belong to f of c. Because we have observed that from union you can go to intersection by complements because the class is already closed under complements. So, let us but e and f belonging to f of c means e or e complement finite and f or f complement finite. So, various possibilities arise let us take case one both e and f are finite. In that case that will imply that e union f is finite because union of finite sets is a finite set and that will imply that e union f belongs to f of c cover collection. So, whenever e and f are finite there. So, what is the case two what is other possibility either e or f is not finite. So, let us for the sake of definiteness suppose e is not finite, but e belongs to the class f of c. So, that implies e belongs to f of c means e complement is finite. Now, e complement is finite let us look at now and e is contained in e union f, but that implies that e union f complement is contained in e complement and e complement is finite. So, that implies so is finite. So, that implies e union f complement is finite and hence by definition this implies e union f belongs to f of c. So, we have shown in either case hence whenever two sets e and f belong to f of c that implies e union f belongs to f of c. So, thus we have shown that the collection so f of c is an algebra. So, f of c is an algebra added. So, f of c which was defined as all sets a contained in x such that a or a complement finite is an algebra and c is contained in f of c. Is it clear why c is contained in f of c? Because for every x belonging to x the singleton x is finite. So, implying the singleton x belongs to f of c. So, implying that c is a subset of f of c. So, thus f of c is an algebra and includes f of c claim finally that f of c is smallest. So, that is let a be any algebra such that c is inside a then that should imply so that is the question that should imply that a includes f of c. So, let us show how is that true. So, to prove this let a belong to f of c. So, two possibilities one a is finite. So, let us write a as x 1, x 2, some x n. That means this is equal to union of singletons x 1, x i, i equal to 1 to n and each singleton x i is an element of class c and c is inside f of c is inside a, c is given to be inside a. So, this implies that implies that a belongs to c. So, let me just go through again because a is finite, a is written as x 1, x 2, x n. So, I can write as a finite union each x i belongs to c. So, each x i belongs to f of c. So, this implies here is because each x i belongs to c which is subset of f of c. So, each one is in f of c, f of c is an algebra. So, a belongs to f of c. This is not what we wanted to prove. So, this is not true. This is of course I am proving whatever was required. So, let a be finite. So, this is we want to show that a is inside the class a. So, each x i belong to c, c is contained in a. That means each x i belongs to a and a is an algebra. So, that implies that the union, so a belongs to a. Same argument basically using that c is inside a and a is an algebra. So, that will prove that a is if a belongs to f, then this implies that a belongs to a. So, that proves that a is a subset, a a always includes f of c. So, hence what we have shown is the following that if c is the collection of all singleton sets x belonging to x, then the algebra generated by it is nothing but all subsets a in x say that a or a complement is finite. So, this is the algebra generated by c. So, we have computed, we have described the algebra generated by a collection c of singletons of a set x. In general, it may not be possible to give a description of the algebra generated by a collection of subsets of a set x. In a special case, when the starting collection c is at least and semi-algebra of subsets of x, it is possible to describe the algebra generated by it and that is our next theorem. So, we want to describe the algebra generated by a semi-algebra. The theorem states the following. Let c be any semi-algebra of subsets of a set x, then f of c the algebra generated by c is given by the following collection. So, it is the collection of all subsets e in x such that e can be written as a union of sets finite number of sets c i's i say 1 to n with each one of the elements c i is in the collection in the semi-algebra c and these c i's are pair wise disjoint. So, what we are claiming is that the algebra generated by a semi-algebra is nothing but the finite disjoint union of elements of c. We have seen an illustration of this in the previous lecture when we described c was the collection of all intervals and we took f of c the finite disjoint union of intervals and showed that that was an algebra. So, that is a typical model example or illustration of describing the algebra generated by a semi-algebra. Let us prove this that the algebra generated by the semi-algebra is nothing but as described above. .. So, given c a semi-algebra f of c the algebra generated by c is equal to all sets such that I can write e as a finite disjoint union of elements belonging c i's belonging to c and note this square bracket square union means that the sets involved are pair wise disjoint. So, let us observe first a few things first of all c is a subset of f of c that should be obvious because every set in c is union of itself single. So, e is has representation as union of itself. So, only one set is involved so e is a subset of and that implies that the empty set and the whole space belong to f of c. So, as a consequence it implies the first property required for f of c to be an algebra and generated we will see it later. So, second thing we are trying to show that f of c is an algebra. So, first property we have checked let us take a set e belonging to f of c that means that implies that e can be written as a finite disjoint union c i equal to 1 to n where c i's belong to f of c. But recall saying that c i's belong to c sorry recall saying that c i's belong to c and c is a semi algebra this is important implies by the property of the semi algebra that each c i can be written as c i complement can be written as a disjoint union of elements of c again. So, let us write it as a i j j equal to 1 to some k i where a i j's belong to c again and they are disjoint. So, this together with the representation for this implies. So, look at e is a union of c i's each c i complement is equal to this. So, that means first of all e complement can be written as intersection of i equal to 1 to n of c i complement and each c i complement is a finite disjoint union. So, this is i equal to 1 to n of finite disjoint union j equal to 1 to k i of a i j. So, here we have used the property namely that e complement by demarcola's is intersection of c i's because e is union of c i's complement and each c i complement being an element of the semi algebra can be written as a disjoint union of elements a j i. So, that is the representation. Now, we use the fact that intersection distributes over union. So, what kind of this set is this can be written as a union of finite number of sets which will involve a i j. So, sets of the following type intersection a k l because when I distribute it over I will be getting one set a i j and some other set a k l and this intersection of that and unions of those and these unions. So, these sets belong each one of them belong to c. Why does it belong to c? Because the collection c is a semi algebra both of them are elements of the semi algebra. So, this intersection is element of the semi algebra and this sets are pair wise disjoint because if i k j and l if any one of the pairs is different then those sets will be disjoint. So, I can say this is a disjoint union of sets of the following type where i belong between one and k i k between one and some k some index and j and l between one and l. So, basically we are saying is e complement can be represented again as a finite disjoint union of elements of c again and that is following because of the fact that e complement which is a union of sets is a intersection and that is intersection of distributive property and this. So, it is because the index is involved are so many it is difficult to write all these things, but it should be clear that e complement is a finite disjointed union of elements of c. So, that means e belonging to f of c implies e complement belong to f of c and let us look at the third property namely if e and f belong to f of c does this imply that e intersection f belong to f of c.  So, that implies so let us conclude from here. So, e intersection f will be equal to this union a i's intersection of union b j's i equal to 1 to n j equal to 1 to m and once again using the distributive property we got this intersection of a i's intersection b j i n j. So, the intersection and each one of them belongs to c, this is union over i and j. So, this by the distributive property will be union over i and j of a i intersection b j and these elements are elements of c because c is a semi algebra a i belong to c b j belong to c and these are again disjoint because they are a i and b j. For disjoint pairs, for different pairs they will be disjoint by the property that these unions are disjoint. So, implies that e intersection f also belongs to f of c. So, what we have shown is the following namely if we take, if I look at c is any and is algebra if I look at this collection of then subsets which are finite disjoint unions then this includes c and this is an algebra. To show that it is algebra generated we have only to show that if a is an algebra and a includes c that should imply a includes f of c. To prove that is obvious because let us take let e belong to f of c but that will imply that e is a disjoint union of elements a i where a i belong to c and c is a subset of a. So, c is inside a so that implies that e is a union of some elements in a and a is a algebra that means a belongs to a e belongs to a. So, hence f of c is a subset of a. So, what we have shown is that this collection f of c of finite disjoint unions of elements of the same algebra is indeed the algebra generated by the semi-algebra. So, we are able to describe completely the algebra generated by a semi-algebra and note one thing we have described the algebra generated by a semi-algebra explicitly and as remarked earlier that the algebra generated by a semi-algebra is described but in general the algebra generated by any collection of subsets we cannot describe it explicitly one may not be able to say what are the elements of that algebra generated by collection of subsets. So, that is not possible always it is only in the case when it is a semi-algebra c we start with we are able to describe the algebra generated by it. So, we have looked at a semi-algebra we have looked at an algebra of collection of subsets of x and then we have looked at the algebra generated by a semi-algebra of subsets of x. Next we go to the next level of collection of subsets which are slightly stronger and that is called the semi-sigma algebra of subsets of x. But before that probably let us look at another property namely how do we generate more algebras or sigma algebras out of a given collection. So, let us take c any collection of subsets of a set x and let e be a fixed set in x. Let us write c intersection e to be the collection of sets of the type c intersection e c belonging to c. Note that c intersection e c is a collection of sets and e is a set. So, c intersection e is just a notation for all sets of the type c intersection e and note these are all subsets of the given set x. So, the claim is that given any collection c and given any set e let us look at c intersection e and generate the algebra by this collection of subsets of. So, f of c intersection e is same as the algebra generated by c intersected with e. So, this is a very useful thing of restricting in some sense restricting the sigma algebras because restricting the algebras. So, what we are saying is c any collection of subsets of a set x e contained in x fixed define c intersection e to be c intersection e where c belongs to c. So, what are these sets? So, note that it is c intersection e. So, this is a subset of, so this is a collection of subsets of the set e and now I can look at f of c intersection e. So, what will be f of c intersection e? It is the algebra of subsets of e generated by c intersection e generated by the collection c intersection e and the claim is that f of c intersection e is same as you first generate the algebra by c. So, this will be the algebra generated by the collection of subsets in c. This is subsets of x. So, this is algebra of subsets of x take its restrictions to e and that is same as this. So, this is the theorem or this is the result that we say is true. So, let us prove this. So, to prove this let us first observe note, so first of all we observe that c is contained in f of c. Given any collection c of subsets of a set x, f of c is algebra generated by c. So, by the very definition c is a subset of f of c. So, that implies that c intersection e is a subset of f of c intersection e. Is that clear? So, if I take a set in c intersection e that is going to look like here c intersection e and c belongs to e. So, that is also an element in f of c. So, it is f of c intersection e. So, this is, so let me write because if a belongs to c intersection e implies a intersection e belongs to c which is in f of c. So, and a belongs to and a is an element in c. So, c intersection e. So, a belongs that means a belongs to c. So, this is in c. So, that is in f of c. So, implies this is an element of f of c intersected with a e and that is precisely the meaning of this. So, we have got c intersection e because of this. This is true and now let us observe that f of c intersection e is an algebra of subsets of e. This is an algebra of subsets of e. Obviously, empty set belongs to this because empty set can be written as empty set intersection e. So, that belongs to f of c intersection e and second. Now, note these are subsets of e. So, what is the whole space? That is e. So, e is equal to e intersection e. So, that is x. So, that is x intersection e and that belongs to f of c intersection e. So, I can write e as x intersection e, x in element of f of c and e is here. So, that belongs to f of c. So, the empty set and the whole space that is e, they belong to this collection. What is the second observation? So, let us look at the second observation from here. Let us take two sets. So, let us take two sets a and b belonging to f of c intersection e. That means this a is equal to… So, if a belongs to that means c intersection e, that means a can be written as some set, let us write g intersection e where g belongs to g belongs to looking at f of c and b is also in this. So, b is written as some h intersection e where h belongs to f of c. So, that implies a intersection b is written as g intersection h intersection e. Now, g and h both belong to f of c. So, this belongs to f of c and that is in a. So, implies a intersection b belongs to f of c intersection of e. So, and finally, let us look at the third. If a belongs to f of c intersection e, that means a can be written as some g intersection e where g belongs to f of c. So, now let us look at a complement, but keep in mind the a complement is in e. So, this is nothing but g complement intersection e. See, this complement is in e. We are taking the complement inside set e because we are looking at the collection of subsets of e. So, this belongs to f of c intersection e. So, what we have shown is that c intersection e is subset of this and this collection is an algebra. So, that implies the property. So, this implies, so let us observe. So, this implies that the algebra generated by c intersection e is a subset of f of c intersection e because this collection is inside this and this is an algebra. So, the algebra generated being the smallest must come inside. So, we get this property. Now, we have to prove the other way in equality. So, claim that f of c intersection e is a subset of f of c intersection e. So, this is what we have to prove. So, to prove this, let us define a collection a. To be all sets, a contained in x such that a intersection e belongs to f of c intersection e. Let us look at this collection a. Note, c is contained in a. That is obvious because if I take an element in c, then that is at a intersection e belongs to c intersection e which is inside this because a belonging to c implies a intersection e belongs to c intersection e which is inside f of c intersection e. So, c is inside a. So, if a is an algebra, what will this imply? c is inside a, a is an algebra that will imply f of c is inside a. But what is the meaning of f of c is inside a? That means for every element f belonging to f of c, if we look at f intersection e, that is belonging to f of c intersection e. So, that means that f of c intersection e is f of c intersection e is a subset of f of c intersection e. So, to complete the proof, we only have to show that a is an algebra of subsets of x. Now, let us look at this and show this is an algebra of subsets of this at x. So, let us start observing. So, we want to show this is an algebra of subsets of a set x. So, to show that, let us observe one. Empty set belongs to a because empty set intersection e is empty set which belongs to f of c intersection e. The whole space x belongs to a because the whole space intersection with e which is e that belongs to f of c intersection e. Keep in mind this is the algebra of subsets of e generated by this collection. Secondly, let us take a set e belonging to a that implies, let us take a set g belonging to a that means g intersection e belong to f of c intersection e and this is an algebra of subsets of e. So, this will imply that g complement intersection e also belongs to f of c intersection e because it is an algebra of subsets of e. So, its complement should also be inside that and that implies that g complement belongs to a. So, g belonging to a implies g complement belongs to a. Finally, let us conclude that if g and h belong to a that imply that g intersection e and h intersection e belong to f of c intersection e and that implies g intersection h intersection e belongs to f of c intersection e. Once again by the fact that this is an algebra f of c intersection e. Once again the fact this is an algebra. So, the collection a of subsets of x is an algebra includes c. So, that must include f of c and that proves the theorem that the algebra generated by a collection c that is f of c when restricted to a the set e is same as first restrict the collection by which you are generating and then restrict and generate. So, what we are saying is given a collection c of subsets of x if we restrict the class c to the subsets of e and then generate the algebra of subsets of e that is same as first generating the algebra and then restricting it that class of sets to that of e. This is going to be very useful later on when we want to restrict collection of sets to subsets of it. .. So, let us just conclude what we have done today. We started with the recalling what is an algebra what is a semi algebra what is a algebra and then we started by observing that in general if a collection c of subsets of a set x is not an algebra we can always generate an algebra out of it. That means we can show the existence of a smallest algebra of subsets of the set x which includes this collection c and basically that is the proof is by showing that if I take the intersection of all the algebras which includes c then that is the smallest one and that also had the important property namely observation namely intersections of algebras is again an algebra. So, that was a crucial property a crucial observation that helped us to prove that the intersection of all the algebras that includes c are also is also a algebra and that includes because the intersection so it has to be smallest and hence that is the algebra generated by. .. And then we gave examples of how to find algebra generated by a collection for example if you take singleton sets of any collection of for any set x then the algebra generated is the collection of all sets which are either finite or the complements are finite and in general if c is a semi algebra then the algebra generated by it is nothing but the collection of all finite this joint unions of elements of that semi algebra. So, thank you so we will continue next time.


So Vinny is stuck up there if George touches his box. No One way to get him unstuck is to literally touch his box and he discovered a new box because a new package came Now Vinny will totally take care of Rocky I have all bets on this guy right here Where are you going?  Hey guys, I thought I would just show you like a Sunday Okay, literally Unedited it's too hot for her sweater. Don't be alarmed for those of you just watching She's just He's cheering because if Rocky moves at all he gets excited This is me trying to edit Vinny is up there. He's about to do another dive What not cases these guys are are you in that case now for those of you interested in bird behavior? He's up there because he's literally getting a bird's-eye view of his boxes because he wants to dive bomb them again We put George over there and then Vinny loves to dive bomb at him He's gonna kill you Press to the off button when I was turning the camera, but here's Vinny He's doing his gang saline.  I accidentally turned it off.  She just screams every time a bird flies Really excited a flock mate. So like they flock together. So well Jersey can't fly but She gets excited. This is what I mean by rules to grudge. By the way, this is just a bit of fun If Vinny's a really good bird, we're just playing around with him Just an angry bird in a Defeated human Being okay, let's go put up flyers for Picasso


The interest that I have on matters of climate change allows for public finance issues because there is a lot of climate financing. Kenya is a country that is basically dependent on agriculture. 80% of our populations depend on agriculture. And for that, agriculture becomes a critical area in the survival of this country. Apart from agriculture, we have other areas that have also been a major contributor to our revenue that depends on climate sensitive activities, that is water and also tourism. And so therefore we need to see how we can actually be able to mitigate the effects of climate change on these critical sectors that are the backbone of our economy. So we developed the curriculum, we developed the training materials, we engaged very experienced people from the Ministry of Environment, the Climate Change Directorate, the other relevant department, Kenya Meteorological Department, and also the National Treasury Climate Change Unit. And with that we formed a team of facilitators. The facilitators that we are using are not our normal facilitators as a school. Because as a school we have not really been having a history on matters of climate change. In fact we have just introduced one very recently. So with that pool of facilitators then you are able now to move out and we did a training in our campuses. The first two cohorts were trained here at Kabete. We had two cohorts trained in Mombasa. We had also two other cohorts trained in Matuga and we had the last cohort in Embu. So for us the biggest output we expected from that training, from that program was actually to reach to the Climate Change Coordination Unit and be able to deliver that message because we had realized that there was a major gap in terms of communication flow. And we have officers who are in climate sensitive departments but do not understand, especially the financing aspect in terms of climate change financing. We have a climate change unit. That unit was created after that program and our director general is very passionate about climate change. So immediately a unit was created and would handle all Mataza environment and climate change. My name is Sheila Mora Ngare. I work with the Climate Change Directorate of the Ministry of Environment. I was part of the training with UNCC Land. The reason why I decided to join the diplomacy training is because I really wanted to understand what happens in the COP Conference of Parties conference. To me before it used to be this mystical grouping where people will come together from all walks of life the whole world and they will all come together and discuss but I didn't understand what were they discussing about how do they discuss, how do they reach agreement on anything. And during that time we learned a lot about climate change diplomacy, what happens in the conference of parties negotiations and everything to do with climate change in general. They were able to provide us with that kind of scientific knowledge of what is climate change, what can we do about it in terms of mitigating or through adaptation techniques, what can we do about it. Climate change does impact the water sector, the agricultural sector, the private sector. It links all of us together. I'm very happy that I was involved in the NDC update in Kenya where at least I gave a contribution towards the youth sector on how the youth can contribute to how we can fight climate change in Kenya and I'm also very happy that I gave numerous presentations to one to the Ministry of Energy to some youth groups at the Council of Governors. I've given numerous presentations or talks to the Ministry of Health. Being a young person I'm able to engage with the youth in their own language so that they're able to understand the climate change technical knowledge in their own capacity. So what I would change at the COP negotiations I would want women of color to participate in the negotiations because we are the ones who are mostly affected, we are the ones who are living in regions that are heavily impacted by climate change but I don't see those faces there. Especially black women. I've not seen any prominent black women in that high level space. To me it seems only women from developed countries occupying that space and I think we need better representation. Whatever is happening at COP whatever discussions are there they should be aired for everybody and everyone to see. It should be running through new stations across the world. I encourage everyone to visit the UNCCLAD website where you'll find numerous courses that deal with climate change how you mitigate climate change how you adapt to climate change. All these courses are free. If you're very passionate about the environment if you know that climate change actually affects you and you understand how it affects you I would recommend this training but also for those who don't understand what is climate change? Is this something in the air that you know is affecting me and I don't know what it is? I would recommend it to everybody. My name is Victor Mugo I am the country coordinator for the climate smart agriculture youth network. The first thing that I really liked about this learning strategy was the recognition that it is meant for especially young people that it was quite great for me to be there as a young person really helping to co-create solutions with young people in the room. I really liked the fact that there were the highest people in the land, PSAs, CSAs, Ministry of Environment, Ministry of Education and the likes. There are common people there are persons living with disability who are there so I really liked that co-creation of solutions together with everyone in the room. It's good to have the people who this policy is meant for and also helping to create that policy and so I really liked that about this strategy formulation process. Whether we like it or not we are being affected by climate change and it has moved from an abstract topic to something that we can see on our everyday basis. At the moment I can see about 1.4 million people who are facing hanka and drought. Then I can also see things such as the rising of the lakes in the Great Drift Valley. Now we see also the locust crisis and so I think it is quite vital for all of us to know what we are facing and I think information is really power especially for young people it's not just information it's insight in how they can be able to undertake activities and programs that can be able to help them to cope fast to the changing climate but also to see how we can hold there are very many numerous opportunities within the agricultural space in which I think young people can be able to tap in. I think we can use the youth demographic as a good conduit for which we can spark limit action in Kenya. I think the first stage I was getting this strategy in place but the other greater thing that we need to do is to make sure that it is implemented and it comes into into fusion in our spaces. My name is Victor Buire. I work at the media accounts of Kenya the premier national institution that regulates the standards and sets standards for journalists and obviously approves courses for any specialized training for journalists. Media accounts and partners had specific deliverables to be specific there were four though we added two or three by way of the project itself giving results that were not initially intended so one was we do we set up a technical committee to look at the gaps in the training of curriculum in the region so we set up a committee that did a gap analysis so the gap analysis was done in Kenya Ethiopia and Uganda to establish what is existing in terms of climate change reporting and we did establish nearly 16 gaps in terms of training we felt there was no any journalism college actually in the region that was offering effectively on climate change that's how our training and especially our relationship with FWA the government of Kenya through treasury and the current nap preparedness training and curriculum came in we developed the curriculum itself because part of the output was to develop a curriculum after doing all these things so we have a curriculum through a competitive process picked 12 people who have been trained first in training or any other person who wants to train on climate change we already have a group of people who have been trained then we had a training of 60 journalists so we trained 60 journalists from 12 counties these are counties that were under reviewed by the Ministry of Forestry and Environment that seem to be vulnerable in terms of climate change the nap readiness program is a huge program what we are doing is a component of a huge project and the UN family and part of it is being done by the Kenya private sector the other bit is being done by the Kenya school of government so media council was picked on the part that relates the training of journalists public communication officers on the issue of creation of awareness through media by creating a pool of journalists and communication officers drawn from government public institutions and journalists from the media to help in changing the behavior because we are looking at the knowledge attitude and practice change among people and media is a key component in terms of setting the agenda and perception about because some of the issues we are talking about in climate change are largely cultural traditional things that people have refused to move and if we don't attack their behavior then the patterns through behavior change using media especially community media radios to help people change perspectives change the new way of looking things then we are failing so the bit media council and the four partners I have mentioned is largely on public awareness especially targeting media and public communication but again like I mentioned to ensure sustainability we are working with the universities and colleges that teach journalism that are done by journalists but there are also now aspects that are done by experts from the university from the ministry from the NGOs to bring in now to give the journalists the content when you are writing a story about climate change for example people invading wetlands what international convention do you indicate to show that there was a human rights violation here now we are talking about the issue of solution based journalism and for journalists to do solution based on context journalism it depends on the understanding of what they are doing


So now we put the whole thing together. BERT's got a couple of different flavors. The BERT base, the standard one that lots of people use, or the Roberta equivalent of it, has in it something that takes in a source sentence, embeds all the words in it, a target sentence, embeds all the words in it, adds in the traditional coding. It then uses a 12 attention head model that pays attention in 12 different ways to the embedding, especially ones here, yet then has 12 layers. So it's a deep neural net goes through and takes each of these attentions, adds them, normalizes them, combines them, gets an overall encoding, does the same thing on the output, the target. But the target has now been masked. 15% of the words in the target were hidden. It takes all this piece, it does a self-intention, it does an encoding to do that. It then takes the encoding from the encoder, the encoding from the decoder, feeds them into another block of multiple attention heads, passes it to another deep learning, and in the end makes a prediction to predict the masked words. And then you train the whole thing by stochastic gradient descent. And the nice basic one has 110 million parameters, but if you have a bigger computer, you can do one that's got 340 million parameters. Ready to download. Cool. You will get to use these prefab in the assignment now and the homework you can try and assemble these things yourself. These are all components you've known and used. You just glue them all together and run the stochastic gradient descent.


My name is David Bederman and I am the vice president for government affairs and general counsel for the National Waste and Recycling Association. We represent the private sector waste and recycling industry in the United States. We have 800 members that operate in all 50 states and we're helping to bring renewable energy to American businesses and residents. Our members have landfills and recycling facilities. They operate waste to energy facilities and they provide residential and commercial trash collection and recycling service. Well, recycling waste preserves valuable landfill space. It helps protect our natural resources and it importantly results in less energy being used to extract new material. So recycling and making sure that waste goes to the right place is really critically important for both public health and the environment.


Welcome to WSIS Forum 2018. I am very pleased this afternoon to be joining Mr. Ali Alomari, who is the Safe City Project Executive Director at the Ministry of Interior at the United Arab Emirates. Thank you very much for joining us. Thank you for having us. So what can you tell us about the Safe City Project? This was, let's say, it's an initiative from our UAE leaders. They have decided to make sure the United Arab Emirates is the safest country in the world. So they have launched an agenda called 2021, and one of these agendas are for security and safety. The Ministry of Interior have put a lot of focus, along with all the local and international stakeholders, to make sure we can achieve this. It's a very difficult, let's say, goal. But if you look at the last, only the last two years, for example Abu Dhabi, the capital of UAE, was in different, in various lists, the safest city in the world. We have took so many different initiatives and programs within the Ministry of Interior, from control centers and building advanced technologies within the control centers, training officers in learning faster and better all the methodologies and transfer knowledge, as well as better fleet engagement and different, let's say, collaboration among various entities in the field. What's interesting about this project or initiative is the fact that it's totally aligned with the sustainable development goals? Of course. So security, even if you reach to the maximum and the safest level, the difficult part is sustainable security. And that's why the UAE government makes sure they are in line of the trend, and as well as exceeding the expectations and keeping it sustained where it becomes the international benchmark. And it's also part of the local government's initiative to create, obviously, a safe and, therefore, happy society? Yes, of course, because with safety then you can do so many things. And our most important goal is also happiness. And that would never be a role if safety is another issue among that. So tell me about your participation at WIS is formed this year. You've been able to share best practices, I imagine, with other participants? Absolutely. My colleagues have shared some examples for the accelerators for the vision of 2021. And we have also shared one of the accelerators related to quick response for emergency calls. So we have different developments using artificial intelligence in our core software where we will be able to, let's say, respond to incidents in a faster mechanism than we used to do before. And this is being continuously improved and developed. So you are already using very innovative technology in order to create this happy and and smart society? Absolutely. Among the Ministry of Interior and all other different sectors in the UAE government. Okay. Well, thank you very much. Thank you very much.


You have an envelope and in the envelope there are three different puzzles Would you take out the four pieces that go that will make this square and there's two puzzles that I would like you to attempt right now All right number one is Can you turn these four pieces from a square into a rectangle that is not a square can you arrange the four Pieces into a rectangle. That's not a square. I'm going to give you the answer to puzzle number one right now But puzzle number two Is a curiosity There's the there's the five by 13 rectangle that you're trying to make but wait a minute Well, it's not yellow that's true But I gave you a square That was eight by eight that has an area of 64 The rectangle has an area of 65 So where did you get the extra? Where'd you get the extra square? It's a puzzle. It's a puzzle Where did you get the extra square? now This is a puzzle that I've played with for many many many years and It's only recently that I learned about some connections that are Bringing me to talk about this puzzle with you 64 equals 65 There's no use trying one can't believe impossible things. I dare say you haven't had much practice when I was your age I always did it for half an hour a day Why sometimes I've believed as many as six impossible things before breakfast In a little catalog of what's At the University of Texas Austin in the forward teacher Turner out and I just saw this about two days ago When one works with Alice marvelous things happen You should have a eight and a half by eleven piece of paper as well, and then you have this puzzle on the back Mr. Dodson drew this when he was a very young man for his family that went into the family newsletter called mishmash and I will leave this as a challenge for you. Can you get out of the forest? Warren Weaver wrote I wish to show how nonsense kept showing up kept twinkling through his mathematics Here's another maze that he drew My goal and I mean me Stewart. I want to show you how how the mathematics keeps coming through the nonsense. I want to thank Martin Gardner because Martin Gardner for one thing I guess I didn't know he had you know was one of the founders of this of this society I Also want to thank Martin Gardner because last year at the Gathering for Gardner after giving a six-minute version of this talk I'm going through my slides. I'm sitting at the table, and there's a man peeking over my shoulder Who hadn't been at my talk and he says to me? Why do you have all those slides of Dodgson's manuscripts? Who are you? And it was Mark Bernstein Mark Bernstein You know noticed the purple ink on my on my laptop and knew that I had something to do here And we started talking and next thing I know I'm standing up here. I Also need to tell you a little bit about Jerry Slocum if you don't know Jerry Jerry is the founder of the International Puzzle Party When I started doing research on puzzles. I Wrote to him and he invited me to his backyard museum his backyard puzzle museum and at the end of the day He told me that I had earned the right to apply for an invitation to attend the International Puzzle Party and Again that has brought different worlds together for me And if you'll notice he's standing by a door and there's another door to hit the right There were two doors to get into the museum. Both of them were puzzles You have to you have to solve what's on the on the door just to get into the museum Jerry's job as a mechanical engineer was to design cockpits for fighter planes Talk about a puzzle. He told me that in World War one most of the casualties from from aircraft were due to pilot error and So designing a cockpit to minimize that was was his career in the ultimate puzzle to try to save lives I got to thank Chris Morgan for this too because at the International Puzzle Party Chris was giving a talk two years ago about his new book games puzzles and related pieces and He showed this slide from Lewis Carroll's notes And he showed the puzzle that you have in front of you And he said but it wasn't a carol original He didn't invent it himself. So it's not in the book and he went on to the other stuff that was in the book And I'm going wait a minute. I want to know more about that puzzle and so Chris Directed me to some places for me to go look and so thank you Chris One of the things I learned in fact, it's not a carol original Forgive me if I keep going back and forth between Dodson and Carol and sometimes I'm not sure who I'm referring to But before those notes were drawn this puzzle was available commercially a wooden version Not only can you arrange it to make an increase but even the puzzle that you have in your hand? You can also arrange it to decrease the area to 63 We'll get back to this puzzle later. I promise I Found it in 1877 it had showed up in the messenger of mathematics journal. I Found that in 1868 it showed up in a German journal. I Found that in 1858 Sam Lloyd Who is considered the premier puzzle master of the late 1800s? He claimed that he had introduced it in 1858, but Sam Lloyd also had a reputation That That we have found no evidence that he really did it But one thing Sam Lloyd did invent was the get off the earth puzzle So let me show you just what this means the get off the earth puzzle. So I'm going to move to Let's make sure let's move to the document camera so you can see this puzzle okay, and There are 13 men Around the around the globe and if you rotate it Let's see right now. There are right now. I've got a point in the right direction right now They're a 13 if I rotate it this way Now they're a 12 Well This is what makes this puzzle so interesting and I should say that there were millions of these produced They got used for all sorts of advertising purposes in particular one of the more interesting sources of advertising was that Was it got used by William McKinley in the 1896 election where they would put the Republican platform on the back and put this on the front Now what does this imply? Okay, this was at a time when the working man's party and in California their official slogan was the Chinese must go So the Republican Party wasn't coming out explicitly and saying it but by handing this out. Oh my goodness 1774 I find in a book called rational Recreations in which the principles of numbers and natural philosophy are closely and copiously elucidated by a series of easy entertaining interesting experiments I Love the title. I find that The first edition in which came out in 1774 shows a three by ten. I'm really looking on the left It shows a three by ten that when you cut it up into four pieces you get two rectangles That show a gain of eight units of area Well, this was a mistake Actually, if you cut it up what you really get is what shows up in the fourth edition Where you get a four by five and a two by six? You gain two units of area not eight now. What makes this very interesting is that Hooper 1774 to 1794 but at the same time Gio writes a French recreational math book that had the same two mistakes and And Gio's came out ahead of Hooper I think we're seeing some plagiarism happening and Hooper wasn't very it took Hooper four additions to get it fixed So don't took Gio only two additions to get it fixed 1545 Serlio Says a man has a table. It's three. It's it's three feet by ten feet But he wants a table that's four feet by seven feet So let's cut the table on the diagonal into two triangles just like this So we have an area of 30 Let's slide that as so So there's our four by seven table and the two triangles add up to an area of three well What we don't know about this one obviously 30 does not equal 31 But what we don't know is did did Serlio do this intentionally or not this wasn't in a recreational mathematics book Okay, we don't know But yet this is the first This is the first thing this is the earliest reference to this kind of a geometric vanish puzzle Where things where the area seemingly disappears or or appears Lloyd patented this idea 1896 Didn't do him much good Because we have so many puzzles like this and in fact in 1889 Fred Howard patented a version of the puzzle as well I Didn't put this one in your packet because it would have taken too many too much to cut up this many for everybody was Okay, but this is one of the puzzles that's on the back tables And you're welcome to help yourself to what's on the back table and there's scissors if you want to try cutting these up Before you before the end of the day if you choose in this case you have a three by eight and When you cut it up you can make two things you can cut it up and arrange it into a five by five square With or without a hole now without the hole, and that's two puzzles without the hole 24 equals 24 Excuse me with the hole 24 equals 24, but then when you arrange it without the hole, where did it come from? Where did it come from? political Messages have not gone away. Here's the get-off-the-earth puzzle in the mid 1950s in Esquire magazine And I know this is I tried to enlarge it as much as I could but you probably can't read it It says swivel the arrow to northeast count the red Chinese Now swivel to northwest and count again. What happened to the 13th man? Why can't the American government solve its recognition problems this neatly? I'll I will leave that be 2006 I'll read to you and then this is they are quoting Hillary Clinton here One international war criminal always has to look over his shoulder if you think he can't be found. Oh someone been lying That Joker George has conveniently forgotten him, but it doesn't take a crystal ball to see the future So you have George Bush juggling? you have Hillary Clinton with a crystal ball and you have Osama bin Laden and when you rotate it Hillary is holding Bin Laden's head in her hands and George Bush is now juggling for the dog and there's one person gone. We've gone from eight people down to seven Same kind of puzzle Martin Gardner. I think my first introduction to this puzzle was many many many years ago In the gotcha book and the vanishing leprechauns where 15 leprechauns can be rearranged into 14 leprechauns And I thought about giving you that one. It's on the back table But then I saw one on this one. I figured let's let's go with the Cheshire cat Why don't you take it out of the envelope? What color is the Cheshire cat purple? How appropriate? So the question is Which one is the Cheshire cat? So put the three pieces together This I think this makes more sense if you can see it right in front of you instead of up on the screen But I'll do it with you up here on the screen All you have to do is take the two pieces that are on the top and reverse them just reverse them now There's a bit of a clue as to what's going on here when you look at this Because with the leprechaun puzzle The hints that are given it says which one vanishes. Where does he go and When he comes back? Where has he been? So let me ask you that question Which one vanished? It seems pretty obvious. We'd say the one in the middle vanished But then I'm gonna ask the question. Where did he go? So I put a third puzzle in your envelope. Oh But I better warn you this was Duttony was a dude was a contemporary I wonder if if Dodson knew Duttony, you know Chris did they know each other? That's certainly It certainly would make sense that they would know you know, they were certainly contemporaries Now according to Martin Gardner, somebody went to jail for doing this So please don't do this with real money anymore. In fact the government has has chosen to counter the counterfeiters You have nine hundred dollar bills They're already cut have you found them? I think they're the I think what color is those those are green of course And so if you cut your money and rearrange them like this well here Let me do this you notice that I've got fractions next to each one of them They all add up to one When we slide it down now we have ten hundred dollar bills Have you ever have you ever had to tape a hundred dollar or a piece of currency back together? It happens.  That happens What does it take to be legal? What does it take to be legal? And it's not more than 50% you don't have to have more than percent more than 50% I have used a subtitle on this talk in the past It's called why us currency has the serial number printed twice on each piece of currency All right, the serial numbers up there twice on each one of them check your bills when you do this rearrangement Yes, you gain an extra bill But those in the middle let's see you don't have to have 50% But you have to have both serial numbers and they better match Okay so can we explain The Cheshire cat and the leprechauns What are these fractions add up to now? so in other words Which hundred dollar bill disappeared? That's a misleading question, isn't it? They all Well, they're all different aren't they? They're all different. It's all been rearranged and it's all hidden very cleverly Here it's pretty clear that they're all a little bit shorter than they used to be in fact They're all one-tenth shorter than they used to be What I love about this I teach mathematics up at Humboldt State University I work with future elementary teachers what I love about this is I can use elementary mathematics to explain this puzzle and To me that's you know, that's why I pursue this. That's where that's why I can make an X why I can make this This is part of my research. This is why I do this is because I'm bringing this stuff to the classroom How can I make mathematics interesting for my students and I'm gonna get back to Dodson on that one When Jerry invited me to his house now, he's got 80,000 puzzles in this museum. He said bring me a new one I haven't seen before I'm going great. How am I gonna do that? Well in the middle of the night The Reverend Dodson was also an insomniac. I gave this But this email that you see on the left was sent to me at five o'clock in the morning Right after I had done this talk at Western Michigan University and the man who sent it to me said I've been thinking about this I've been laying awake at night and You see these six men here Which one disappeared They all did didn't they? And I think that this one really demonstrates what I love about this is it demonstrates the paradox and helps to see through what's going on so Chris pointed me towards the parish collection at Princeton University, and I had the the wonderful Opportunity to spend a day there. I need to go back. I have to spend more time I feel like I'm just beginning Well, you know what I'm doing and you know beginning of what you're seeing here is only the beginning of this I found thousands of pages of mathematical manuscripts in boxes some of it very unorganized actually Some of them just scribbles of computations all in purple ink Some of it was absolutely gorgeous. There's a proof that the of trisecting an angle or Should I say That you know his explanation and his is not done what what he did here was not strictly compass and straight edge for those of you that Know more about this, but he did this when he was 12 years old When I went to parish the parish collection at Princeton Stan Isaacs who I know many of you know And thank you Stan for pointing me and getting me to look for this Stan was looking for a particular billiard ball problem So every time I found one I took a took a photo of it for Stan It says find the path of a billiard ball in a cube Which goes on the same path forever and strikes all six sides gravity neglected At the top of the page it says I thought this partly out and wrote out the result on another occasion Sometimes he would write at the top of the page not thought out Here the goal is to have a billiard ball travel around a non rectangular table Here's a third one Where he says a billiard ball A billiard table is nine feet four inches long And it is Let's see it is kind of read it better. It is nine feet four inches long. It is 10 feet wide And the goal is to it starts in the center and you have to Let's see the goal is to hit all it has to hit 16 cushions It and then hit into a corner pocket and it has to travel a total of 75 feet 10 inches What you see on the right is Dodson's solution to this Some beautiful mathematics here using using reflection symmetry the understanding that that all every one of these tables is just a mirror reflection And but he's laid it all out in one straight line I'd be happy to make these slides available if you're interested and want to get a closer look at these I found a fourth pool table billiard ball problem in the pillow problem in the pillow tales book A triangular billiard table A point is given A point is given by tri-linear coordinates a ball strikes all three sides and returns to the starting point find the Find the point where the ball strikes the second side Again many many problems like this I found these notes. I love this one. Okay, it says on the bottom Can there be traced in one line each or if not in how many Going by the root of always Crossing to a new oval and every point of contact. Basically what he's asking Can you draw this without picking up your pencil? That's your goal. Let me read a letter Or no, this is not a letter But this is what you see on the right Again, the goal on the left is to draw it without picking up your pencil 40 years after let's see After he died let's see one when his nephew stewart collingwood was writing his biography He got a lot of feedbacks from many of dodson's young friends She writes we met for the first time in the forberry gardens He was I believe waiting for a train. I was playing with my brothers and sisters in the gardens I remember his taking me on his knee And showing me puzzles the puzzle this puzzle was by the way a great favorite of his The problem is to draw three interlaced squares without going over the same lines twice or taking the pen off the paper Which is so thoroughly characteristic of him in its quaint manner So two very the two puzzles are very much the same And it made me think about this problem Which in the 1700s the people of konigsberg prussia they have They have two islands Seven bridges that connect the two islands with the mainland And one of the things that they like to do would be go to go for a walk And what they wanted to do was cross all seven bridges Without crossing any of them twice This was the beginning of graph theory And this was the beginning and it was Leonard oiler who was able to prove that it's actually impossible to cross all seven bridges Once without having to retrace your steps And then I learned that when when dodson traveled through europe he went to konigsberg Makes me wonder if he had any interest in this puzzle because he gave children very much a similar puzzle And he went to konigsberg. I have not found any evidence. He never wrote about About this particular puzzle and I haven't found anything out there. Please let me know if you know anything about this one of the You know and and he even took it to three dimensions because here I found in these notes Where he's trying to wrap a string one string all the way around a cube Very much a similar kind of a puzzle here Can you take one piece and go all the way around the cube wrapping it as so And then tucked in between the manuscripts. I found this little two little cards two letters that Mark said you have to share these Stuart and he said you can't not share these Let me read to you Dear mrs. Jarrans. I'm reading the one on the right dear mrs. Jarrans And he wrote this less than a year about just about a year before he died I would have much liked to come to you last night and was quite sorry to be Debarred by the form of your letter which was practically an invitation underscore At present I can tell all friends and truthfully that I never accept invitations and I can't surrender My present secure solitude even for the pleasure of an evening with you Please only send me information in the future and if you add I shall have joy To see you please spell it with a big j Sincerely yours cl dodson There's been a lot of question, you know about his relationships with adult women There has been this letter. I know I'm not the first one to find this because I know that Edward Wakeling included it In the diaries. So there there is some information about it, but Very curious And the fact that what we do know is that as he got towards the end of his life He was shunning any Adult social you know socializing with other adults because he wanted to do more mathematics That was what he wanted to do and he didn't want to give up an evening Although children were another story Where does the day begin? This was an interesting puzzle of his And I'm going to just I'm going to read to you It says if a man could travel around the world so fast that the sun would be always directly above his head And if he were to start traveling at midday on tuesday Then in 24 hours he would return to his original point of departure and would find that the day was now called wednesday At what point in his journey would the day change its name? Well when he wrote this when he first put it out there was no international date line And by the time he Before he died I put it was 1880 something when the international date line was established. So this was a real issue and I don't know if he had much influence in the creation of the international date line But certainly i'm sure that this question was out there It had to have caused some of the interest in the conf you know the confusion here imagine that you know this is certainly something and and That he thought about it and it puzzled him and it bothered him as well Charles Dodson would refuse mail that was addressed to lewis carol. He would deny that he was lewis carol And You don't see much overlap, but here is a page of his notes He writes 17th in decimals Is point zero five eight eight two three five two nine four one one seven six two four seven and it goes on and on and on And then he goes on it's to write a whole lot about this particular circulating decimal But look what says up at the top of the page Look wild no hymns. Oh my Where did that come from? Where did that come from and i'm very curious about this one because as I said carol and dodson were almost two different people But on this page it seems like there's something bringing them together here And what is it about this particular page? You know about this particular one that made him do this He collected all kinds of mechanical devices As a boy he would make puppets he collected music boxes Um, certainly mechanical puzzles. He always had a puzzle in his pocket Okay, that you know that there's the handkerchief mouse that he always had Um, he always had a wire puzzle by the way this little guy is needed to get some air here He needs to get some air and even take a handkerchief and fold it up and do a little pocket mouse okay And Isa bowman No, let's see You know, he had this let's see one of the young girls writes next I went out looking for my twin daughters I find them seated with cld seated between them and they're listening to him open mouth in the greatest state of enjoyment With his knee covered with my new toys Always toys In his diary he writes I met my blue china friend and had some talk with her and tried her with my usual card of introduction The wire puzzle On another day he writes on the railway going back. I made friends with a nice girl of about 15 who was on her way to school in eastborn I gave her my card and promised her a wire puzzle Here's a letter. He writes my dear alice And There were some references that this might be princess alice The queen's daughter Please excuse me if this is written very badly, but how can one attend to one's writing? You know when a great harry green thing is crawling all over the letter And he did draw a picture of a spider on this letter I shouldn't mind it so much if the thing would just keep still What I didn't like most is that it will crawl about all the time. Isn't it provoking of it? Please give this note to your mother and this puzzle to charlie The two wire men are england and ireland and the puzzle is to make them join and unjoin their hands They give us sort of rule for doing it, but even with the rule It's rather hard. Tell them it goes quite easily if you do it right away So he mustn't push hard to get the wire loops over each other And if he does it'll get into such a mess. You'll never get it right again Give charlie my love and take two or three crumbs of it for yourself Yours affectionately now We don't know that this What that puzzle looked like one of the things is he gave away a lot of wire puzzles, but we don't have those puzzles Chris morgan did some research and this was in james dalgettis collection And it certainly seems to fit the description. It certainly seems to fit the description that this might be it He writes he writes this afternoon I shared a bench on the marine parade with a gentleman and his wife and a nice little girl about 10 To whom I showed some puzzles a pleasant child, though not very bright On a sunday afternoon he writes i met mrs. Blakemore and dolly at about one on the parade and dolly not only spoke to me without crying But actually asked me to come in with them stayed about an hour showing her puzzles etc And it really looks as if they were good friends at last after five weeks estrangement Went over to brighton and talked to the girls school 20 girls for whom I did puzzles For more than an hour and a half Isa bowman wrote about how he would keep things in his apartment And so so oh my goodness Bob the bat well, this is actually this is the closest I could find to bob the bat and He would fly about that children loved it He kept it in his left-hand drawer What would he do when uncle wound him up? He would really fly Those of you that are engineers, you know, here's a man who's an insomniac He's always thinking about things and anyone who has tried as I often have done the process of getting out of bed at 2 a.m In a winter night lighting a candle and recording some happy thought which would probably be otherwise forgotten Will agree with me it entails much discomfort Now all I have to do Is to draw from under the pillow a small memorandum book containing my nictograph Write a few lines or a few pages and without even taking it outside the bed clothes Replace the book and go to sleep again and here's his nictograph that he invented And this is the code. This is what it would what it would look like when he would write So he was designing, you know, so clever so clever for those long nights And I'm also now I'm here I'm going to start watching the clock and go through a few of these and I need to say a little bit James Newman wrote that he was a mediocre mathematician without brightening the hour of a single student Or producing anything of lasting value to his subject Warren Reaver wrote in all of dodson's mathematical writing It's evident. He was not an important mathematician Jenny wolf wrote he was too focused on rules and procedures. He was unattractive to his students A student wrote he was singularly dry and his perfunctory manner in which he imparted instruction to us never betraying The slightest personal interest in matters that were of deep concern to us As I explore his diaries as I read from other people, I'm finding that this wasn't true Or maybe what I'm finding is that he was misplaced. He should have been an elementary school teacher And not a university lecturer Because when he after he retired he started to go to schools to work with children and they loved him They absolutely loved him. They would look forward to him coming Little girls, let's see You know, his fancy was sometimes suffered to peep out little girls would learn the rudiments of calculation at his knee They found the path that they had imagined imagined. So thorny set about with rosin's by reason of the delightful fun Which which he would turn task a task into a joy When the fun was over the little girl would find she'd learned the lesson just the same When I went up to oxford I learned from mr. Dodge and to look upon my mathematics is the most delightful of all my studies His lectures were never dry. So we can find both arguments even at oxford Do you ever play at games or is your idea of life breakfast lessons dinner lessons tea lessons bed lessons breakfast lessons It is a very neat plan of life and almost as interesting as being a sewing machine or a coffee grinder But it seems like that's the way he taught at least at oxford. At least that's what most of what we see But then here he is teaching games again Where one player names three or four letters in a word and the other has to guess the word many many word games like that and he one time he went To a lot he writes in his diary that he gave his first logic lecture at st. Hughes hall to 13 girls Including eval and hatch and eval and wrote 40 years later That my old friend mr. Dodson offered to come and give us a lecture on logic With great eagerness my fellow students prepared to meet the famous mathematical tutor Who was the author of alice in wonderland and assembled in the library armed with notebooks and pencils to their surprise The lecturer appeared with a large black handbag From which he proceeded to draw a number of white envelopes to be distributed among the audience Why are you laughing? Each envelope proved to contain a card marked with two square diagrams and nine counters some pink and some gray Notebooks and pencils were not required. We were to play a game. He turned logic into a game Stephanie you made a reference a couple hours ago about a venn diagram happening up here And i'm just beginning to discover that as his if he had lived longer He developed a something and similar to venn diagrams, which we will call carol diagrams And what i have been reading and learning and i need to go learn more is that if he had lived longer It's possible that he would be remembered as a logician and one who did contribute to the field of logic Because he did do quite a lot with it And it is said that these carol diagrams actually work much better With larger sets and again i'm not going to get into the details I'm not even comfortable with it myself. I'm just getting into it, but You know, but yes We had a venn diagram up here, but we also had a carol diagram up here as well In the preface to symbolic logic he writes It's used mostly in enabling one to asserts one mental one's mental powers It gives the clearness to see your way through a puzzle the habit of arranging your ideas in an orderly and get Addible form and the power to detect detect fallacies and other people's arguments In the preface to curiosa mathematica he wrote it may well be doubted whether in all the range of science There is any field so fascinating to the explorer so rich and hidden treasures so fruitful In delightful surprises as that of pure mathematics The charm lives chiefly i think in the absolute certainty of its results for that is what beyond almost all mental treasures The human intellect craves let us only be sure of something And i think that you know it shows not just a love for the topic But again i find that he has a way with words that i can't seem to get across to my students This love of mathematics just for its own sake, but he puts it in words here that makes me think You know he he's he could do it. He could do it and but he was in the wrong place and perhaps at the wrong time He loved nothing more than to invite children over to his apartment for tea and logic An excerpt an excerpt from his diary He loved number tricks And again many many many number tricks about who can get to a hundred first He loved numbers my dear guttured. I send you seven kisses to last a week Your loving friend my dear guttured gertrude. I send you 10 million kisses and remain your loving friend see al-dadsin My dear guttured gertrude. I send you four and three-quarter kisses My dear gertrude This will not do you know sending one more kiss every time by post the parcel gets so heavy It is quite expensive when the postman brought in the last letter. He looked quite grave Mind you don't get any more such letters. He said at least not from that particular little girl I promised him we would send each other very few more letters only 2470 or so Oh, he said a little number like that doesn't signify what I meant is you mustn't send many So you see gertrude. We must keep count now and when we get to 2470 we must not write anymore Your loving friend In this little letter here, he talks about morning dress affairs as opposed to evening dress affairs His dinner parties are morning dress affairs and he also writes at the bottom. I will come for you at six and a quarter fractions used to tell time an interesting way to do it or if If each if in picking a coral if each party declined to go more than three eighths of the way And if in making friends each was ready to go five eighths of the way Why we would have more reconciliations of quarrels I love it elementary mathematics To to reconcile The consumption of mid-jera has been during the past year zero After careful calculation I estimate that if this rate of consumption be steadily maintained our present stock will last us an infinite number of years And although there may be something monotonous and dreary in the prospect of such vast cycles We may yet cheer ourselves with the thought of how economically it can be done Dividing by zero. What a great way to just to to demonstrate it. I think I will Go through i'm going to watch because I certainly want to share this one My darling isa it's all very well for you and nelly and emcee to unite in millions of hugs and kisses But consider the time it would occupy your poor old very busy uncle Try hugging and kissing emcee for a minute by the watch and I don't think you'll manage it more than 20 times a minute Millions must be 2 million at least and then he goes 2 million divided by 20 2 million hugs and kisses divided by 20. That's 100 000 minutes divided by 60. That's 1666 hours divided by 12 hours in a day That's 138 days divided by six days a week. That's 23 weeks I wouldn't go on and any rites. I wouldn't go on hugging and kissing more than 12 hours a day And I wouldn't like to spend Sundays that way So you see it would take 23 weeks of hard work. Really. I can't spare the time I struggled to get my students to comprehend what a number like 2 million means Here Dodson is doing it back when supposedly the elementary curriculum was very dull And he's finding ways to bring it alive. He's finding ways to make them interested and make it connected He came up with rules for addition and subtraction and rules for for division He came he explored cyclic numbers Which all I want to say about repeating decimals that go on and on and on in a repeating pattern and all I'm going to say about it is that In his eight or nine wise words about letter writing. He wrote Don't repeat yourself When once you have said your say fully and clearly on a certain point and have failed to convince your friend Drop the subject to repeat your arguments all over again will simply lead to his doing the same And so you will go on like a circulating decimal Did you ever know a circulating decimal to come to an end? Again, he brings math into some really unrelated places Here's a notebook with his with his lecture notes for algebraic geometry Here's the pages I was looking for obviously I found a lot more I'm going to spare you the details that are on this page, but I will say this If you rearrange those four pieces You're going to notice a long skinny gap. Guess what the area of that gap is right in the center and what he did Is he got into exploring where we gain? He came up with some methods.  So he Came up with some generalizations about this puzzle that I wasn't aware of Notice in this diagram, he's emphasizing he's exaggerating this gap And what's interesting here is he's explaining it in a way that I hadn't explained it before I hadn't I hadn't used his explanation He's also using notation that I hadn't seen before as well He found he he found the area of that gap using a formula that I wasn't aware of either You know about doublets So I'm going to go on with I'm going to going to keep going with that And I guess what I will do here To finish this up He knew about the 15 puzzle In fact, he spent an evening with the 15 puzzle Which is sort of the the this puzzle Took america by storm And the only thing that's come close to that has been the rivix cube This puzzle is on the back Notice when you rearrange it one rabbit just you know, but it's easy to explain this one because where do rabbits go?  I won't read it to you But you can turn it into a seven line poem if you like And it's on the back table by the way it is on the back table He wrote It occurred to me that if we take letters and put them on a chess board and move them around they might form words Is this the beginning of where scrabble came from? Do I have time to finish with this? Very quickly very quickly I will finish With the martin gardener dollar because you see In from the country of money That if we move martin out of the way, whoops, you can't see it. Let me escape and go here and full screen So here is Here's a martin gardener dollar bill. It's a very special dollar bill Because if we cut it up as so and we remove martin and then we Take the pieces and turn them upside down and put them back together Where'd martin go? Well Let's see we turned martin over Thank you very much


I am not here to fit in. I'm not here to keep you happy or them happy or this happy. I'm here to live out my days at the fullest best evolved version of me I can possibly be.