Merken
Math for Economists  Lecture 14
Automatisierte Medienanalyse
Diese automatischen Videoanalysen setzt das TIBAVPortal ein:
Szenenerkennung — Shot Boundary Detection segmentiert das Video anhand von Bildmerkmalen. Ein daraus erzeugtes visuelles Inhaltsverzeichnis gibt einen schnellen Überblick über den Inhalt des Videos und bietet einen zielgenauen Zugriff.
Texterkennung – Intelligent Character Recognition erfasst, indexiert und macht geschriebene Sprache (zum Beispiel Text auf Folien) durchsuchbar.
Spracherkennung – Speech to Text notiert die gesprochene Sprache im Video in Form eines Transkripts, das durchsuchbar ist.
Bilderkennung – Visual Concept Detection indexiert das Bewegtbild mit fachspezifischen und fächerübergreifenden visuellen Konzepten (zum Beispiel Landschaft, Fassadendetail, technische Zeichnung, Computeranimation oder Vorlesung).
Verschlagwortung – Named Entity Recognition beschreibt die einzelnen Videosegmente mit semantisch verknüpften Sachbegriffen. Synonyme oder Unterbegriffe von eingegebenen Suchbegriffen können dadurch automatisch mitgesucht werden, was die Treffermenge erweitert.
Erkannte Entitäten
Sprachtranskript
00:05
But we made it to the end this is the last lecture of new material facts that presented all the material in a practice some more and then Monday review or give you a bunch a sample of final exam questions on Monday and soldiers go over those who take their final on Wednesday a final is at a very much
00:29
Made it to the end of the last subject here understood a reproduce what I did last time just the general formula an outline for a look Grant multiplier This is a technique will grant a multiplier title here on what it is is it's a technique for finding Max's Indians when you're restricting yourself to just a set of points along a function of a function I'm only allowed the plugin points along little squiggle and then The way we find the maximum in relative to that situation as we use these Look grant multipliers do to the situation is like a described last time we have A function Got a couple variables and by the way this can be done or not doing it for more variables that this can be done with 3 variables and for variables it's the same technique so if you're interested in that you can take a class I like math 2 D order a look in a calculus just a some examples of most calculus books will have OK so our situation if we can build up to this situation from all the other things that we studied American in a review of getting ready for a final so original situation was here's a here's a function of finding the maximum an overall so inhabiting restrictions we don't have a rectangle didn't have little squiggle Following just say go find the maximum in the global maximum and the way we did that was we took the gradient said illegal 0 found a critical point and then we tested that critical point using that Passion and then we found conquered up and down we gotta maximum than the next level was we restrict Taneyville rectangle Bernanke rectangle up there and the registered in the maximum end in their little region The we did a bunch examples of that than the last step that world which is what we're on now if we will don't restrict to a rectangle don't say it's everything possible what we do is restrict to what's called a constraint and that word constraint if you study economics and I don't know how much You guys in a steady but that word comes up a lot in the context of a budget constraints so when I say the word constraint you should think in your mind budget constraint to have some production function some company or something that that produces some products but they have some restraint on how much they can produce and that used to call me but I'll look at some examples of that a little bit of this your constraint And so what I do is am only allowed the plugin points along there's select his below squiggle up there and I'm interested in finding the maximum in clothes can a complicated that way so what we do is we take the level set this function here levels said so that Function and were looking for when the level said is candidates of the budget constraints and you might get a few of them so those 2 points of Tianjin and the significance of the tangent is that they gradients airline that they're in the same direction they might be cut both pointing this way or once pointing this when one's point that way but they're lined up at that point when their Tianjin so if you just know you know if you look at these levels said they represent heights so if you could just can't Just as sample case imagine eases cover ramp going up So this is smaller than this 1 and this was smaller than that 1 so you can see if they're going up this Represents very 1st level where we hit the constraint and so that even minimal if this was going I suspect he at 1 at books and so on so the very first one That's going to Tianjin point that can the minimum and then as it extends Further review more more these levels said the very last 1 hit that maximum so study it that way you realize OK that if I could just fine with those vectors lineup by automatically have the maximum and minimum at locations for Just solving magazine locations in the news that Maybe do a little extra analysis to find out whether you actually Maxtor and we saw that last for all the work is And they just finding out where those gradients lined up so we have named the gradient F is equal to Atlanta times agreed g fat labeled anything yet so it but some labels and fears at this guy is she that's why can't When I write it this way it implies that we're looking at a level said Everlarger functions here's a larger function but then we global constant that gives us a little graph down the XY plane that constraint but hidden behind the scenes is that there's a larger function and that a function that we take the gradient Both men line up with matters automatically finds you the maximum in eyes So just have to learn how said Had a deal with fee out focus so the formula Like I said last time the recipe here 1st all the agreement that People's land times the gradient of GE and when you solve this this You can kind of see it nevertheless sad if I have to variables like this situation than I get that 2 components for the at so that's enough supply GE setbacks and usable I to Equally Asian sitting here but I have 3 variables so at this stage all you can hope for is to get a relationship between X and Y look back at the last 2 examples when I saw that I don't get a number I get a relationship X equals y x equals to wire or something like that and it's that relationship then that you plug into the constraint that we take that information we put an end the constraint and that gives you the actual points These little locations 3 here showed up at the plug results from 1 him to constraint and then plug the results of Plural you make it more than 1 . 5 So what results from 2 but once I get now at this stage I take this relationship between XOY put it into the constraint and get an actual set of points then I take those points and I put them into that's At her and if that doesn't say only a onepoint Delia 1 point you can't tell it's a Maxtor have to plug in some other test point to see whether you had an maximum but apt you've done this process you definitely have the location of the maximum and you just may not have complete information and find out what we have maximum without pointed to a
08:11
Gruesome examples of in of the the thing about this is the methods always the same what happens is as you change the functions namely F And the constraint GET the algebra For solving this step 1 becomes harder and harder And if you were to look into a calculus book the examples they do in those books they think it very hard very quickly so II and is building the up slowly the 1st example was real simple algebra center was slightly harder because we gotta a plus and a minus if you look through their there was this extra point now each 1 leg is in successively hard it With so the woods to look at my view example there's so that should be familiar yeah we've seen his Hovard over again but it's not look it is probably 10 days is notes that function by an OK so this
09:24
Guide write down immediately and know this is the paraboloid And what of the level set that look like they look like little concentric circles down next White Plains that's the equivalent of
09:38
Little curves there
09:41
And then only give you might restraint astronomy a little different protects y Equals 1 peso again it said that's a larger function GMX y He calls exwife write that down and then we're considering just a particular 1 particularly levels that were orgy of x equals 1 So let's let's get the constraint on the board here I want when I grab this usually readers just like that white something to see would draft looks like so If I had X Y equals 1 that implies that have equal there's 1 over X looks like this After months of . 1 1 Now what couldn't hear the level steps of this refuge pretty familiar with The level set up half theories Equals 0 rate there and then if I have a radius of 1 that's here Monday That was 1 and if I go out At most 4 then I get this guy here That s equal for war So somewhere in between is the very first one that I hear I want this 1 right here I want the 1 that Jesse's tangent I'm interested in finding And then so what'll happen as 1st will find get these pointer there were looking for 2 points so that is a little advantage in taking some time to try to dry these graphs because you think you can know While you're looking for here on at I can tell from the picture what my answers Are gonna look like a minute to positive values every year to negative values of the letter were looking for now you didn't have to draw the pictures in order to do this method because it is says all that equation solving quiff follow this recipe for Coca so the gradient of death that's not a mystery at this point there's just 2 X 2 Y and that's going to hold slammed a times G. that OK that what's the school over and do that separately
12:16
Wrote broke Woman racist just right equation 1st gradient of They have was Lambert Gradient of GET the gradient of That is a Q X 2 Y and Lander And the grain of just be careful that doing the derivative of this guy with respect to
12:41
X so I get what a constant for a while I hear the derivative of this with respect to Y X is behaving like the constant But This will set up this gives me to equations I have 2 x equals Lander why and I have Q. Why equals land next to what suggested to use each time is when you're at this stage in order to try to solve this sulfur lamp United need find out what land is anyway I'm looking for X Y Look at this location so use landed And salt for here and then you'll have equations With just Edson wire 1 equation which is 601 argued that relationship that you need so take 1 of these guys this says that landowners equal the Schuette over why eminent take that and put it in there now I have to equals 2 Over times next Choose canceled and basically that with why squared equals X square or Why pools and we take that square if you leave out the cluster manager We get other solution United in this case
14:10
OK so it's continue we just finished the 1st that here we solve this and when we get we get a relationship between X and Y you see this to relationships there's Y equals X and his wife equals negative so now I'm starting on step 2 on plug the results from 1 into the constraint my constraint is X Y equals 1 and my results from Part 1 I have I have 2 things I have X Y equals X and I have a wife negative those 2 relationships need to be plugged into the constraint to find out the actual value
15:00
Focus was do I equals X Y plugin White equals X in here I get squared equals 1 So I get a gold mine 1 It gets a little tedious Erietta keep track of thanks to Niger's get to values of X and in this scenario here why equals X. so when looks just break it up into 2 parts I've got equals 1 will then that implies white equals 1 in this scenario white balls acts and that I had x equals negative 1 and 1 equals X. so that implies y equals negative so from this analysis here alone I got the 2 points I'm looking for actually I get He followed this implies that we have 2 points we have won 1 and A negative 1 made it 1 of the 2 points that solve that equation were done with step to for this Now I have to go over and do the other relationship White was negative X and see what I get for that That's why equals negative excellent plug that in here I get tax times negative X quotas 1 which means negative squared equals 1 what's the problem with that Yeah we yet what let me if you had a consolidated X where equals negative 1 but the point is that this is always negative an equal sum composite solution a solution so this 2nd relationship didn't yield anymore and A new route you realize that from me out of my guess technically if you just didn't even notice it and you never even if the party stood at the ready answer in the next example received the 2nd solution is gonna give us answers You can't just Pretend like it's always gonna happen OK so we got our 2 points we done step to hear the necessity the results from step to employees in Athens the maximum 3 F of that's why next with close twice with so plug in those 2 . These are the locations where the Max's Noonan's Oregon And I'll discover which ones which when I do this at 1 1 that's good 1 square ones where just 2 and half of a negative 1 on his eagled 1 square to so right now but I can tell When I get the same value for each 1 so now what we do what we need to check so I got the same value I can't tell this is Max for I know I have the right locations in either Mac sermon that I can't tell you So the check 1 1 . check members want in this problem here were only interested in points that come from the constraints so when I say check the point I specifically on the constraint and make sure that we're still living there I can't just pick any point on the graph On death OK so that's X Y equals 1 so just take anything about 2 and a half to 1 half of try to wives half 2 1 And what is that You 1 have yet to square 1 half so that's for plus a 4th 5 which is greater than she and some other point on the a constraint has found its larger than those points for those mustard minimum Look right at the Final Solution areas face over this case so what we found was at 1 1 in negative 1 1 those of those 2 points there and you can tell from the picture that their minimum CC because see the user smaller level set the very 1st level said that intersect is the smallest level set intersect as they go to larger level said they do intercepts of the very first one is as 1 below that laughter at sequels to so I have said that that is where we have to act watch 1 lies the minimum value of cursed at 1 1 and negative 1 negative 1 and 2
20:50
This is the minimum value of that subject
21:02
The Any questions about that and stare at their meeting up there Now we have to raise the rest More examples suggested before when you wanna learn something new will you do is you take this example clean visa paper any cheddar reproducing you'll get stuck amid unless you got it already
21:33
That you get stuck in you know is that next step earlier wiser plus or minus something you look at the solution and then you just keep doing so you can reproduce it without looking at you don't move on that's the way you learned that you have to be willing to do that you have that have that Curiosity period he changes force that yourself but if you're willing to do that that's how you learn that they download the for the next 1 will ruin it is resisting switched the roles some dinner make up of x equaled X Y and I'm animate constraint equal to that of the circle so you know we have the same graphs kind of time Look It's so that s that's why That's why ends at my constraints Is going to be squared plus Y squared off more of these 2 appears that's right down the constraint Reddaway with Backdraft looks like this 1 negative 1 OK so there's that the unit circle And then let's go deeper the level set so this this is a little bit more complicated with the level set of this look like let's start with a 1 we've done this 1 before it's worth it to review it so onehalf equals 1 that and we have X Y equals 1 his did this 1 implies that while he was sex so I have 4 That equals 1 I have level said 1 over This is equal to them Let's do it That was negative 1 I I can cheating because I know the answer I I need to pick the level set that's negative so that I can show you the other part of that's so when I do at equals negative 1 and that means I have exwife cause negative 1 instead soon I have twice quiz negative 1 of Texas Negative won over X Everything if it's negative If this was positive memories of the 1 below their access I'm just not applying this graph by negatives all this stuff becomes negative in all the negative stuff becomes part of this of this represents It had negative or that want to positive color in here But emphasized Now That's very equals 1 genetic was negative on the other side wants to a 3rd year due on ethical 0 it as equals 0 then They were just follow the same pattern that if you wanna said X Y equals 1 9 equal 0 will do X Y equals 0 Boodarie have 2 things multiplying together give you 0 1 of a massacre 0 this implies that seat with 0 Or why equals the graph of Mexico 0 that's the Y axis of the graph 1 8 0 0 that's the Xaxis uses 3rd color here the 3rd level said is this guy here As the ax equals 0 their white Sosa a more complicated set of level sets for this particular village put those over here level said that I just did 3 examples of level sets but so here's Yukos 1 way I did that equals too The nite end up with something else out here and The event here if I did it he goes negative to I get something like that so the set of level sets are things like this season the level set and I'm interested in 1 Manchester is Tianjin here UCI winning get several of ideas on over here this these 2 just like the last example but and this particular case we get a couple extras we get the ones over here too I'm look into these points race here and there's no 1 appears well just hangin those are all level set so that I could put this 1 in the middle there but is not that could be relevant to this not the engine look the and so my solution will be these 4 points here know know what they are in advanced by 1 1 in the negative automated 1 Natal and so on but But maybe not afraid it's it's just that there be glad to have to be honest 1 1 is known will not be at 1 point but they interviewed book excellent now down to our little recipe hearings and find 4 points algebraic So step 1 says Ms. Alves mess and the gradient of that In here Limited derivative with respect to x y that's behaving like the a constant lunatic too derivative with respect to why again so that's the gradient of death and that the Lambda times greater G. What's G GE The paraboloid we're looking at that particular level said there's G functions so now we take agree that I get to X Y components 9 at my 2 equations equals to land X and X equals to Lander Wyo It will be consistent here and sulfur land at that time meaning it this little jam in here so 1st things 1st with sulfur land planned equals Hawaii over to X and take that very land there a year and put it in there and I have equals 2 times while 2 x y In that implies school here But his choose Cancel I get just what I had last squared equals 1 squares And here again I can salt from wife in terms of That's a member of the 1st step you don't expect to get your answer you get your number will you get is a relationship between these 2 variables and the reason you expect that because I have 3 variables into equations I had no hope of finding an exact solution until I added a 3rd equation and that's going to be the constraint if with their city have used them OK so I got this relationship of plug it into the constraint that with the the 2nd step said But 2 of these I got Y X Why 2 separate problem white equals X flood that ended the constraint and I get that I have squared squared equals 1 2 x squared was 1 of OPEC's where half of taxes Eagle of plus or minus 1 Overridge too
31:03
But he see That's that's too solutions at X 1 on a route to which implies that why is 1 of original because
31:15
In this scenario equals X and then over a year ahead the other case here sequels a negative 1 Overridge and then that must mean that is equal to negative what So putting them altogether I have 2 points here from this scenario I have the the . 1 over a route to 1 Negative to negative 1 or 2 If you look back over here in this picture but we just found was this
31:44
Point and the 1 in the 1st quarter And the other 2
31:52
Find a really get that From the 2nd scenario White equals negative The 1 equals negative X to plug it into the constraint X squared plus Ganguly Squared equals 1 c here the negative doesn't cause any problems we actually get the exact same solution to act quickly as 1 X squared equals half of the tag sequels Plus or minus 1 over route to this pairing is exactly the same as this but here's where changes we see I have 2 versions of this x 1 over route to that you see in this scenario why name 6 Soviet Equals negative want that's a different point And here I have text equals Negative 1 over too well in this scenario wise equal negative X so that employers Y is equal to the task So over here in this case I got 2 . 1 per room to negative honorary to an idea and negative on a route to positive those of 4 locations where these gradients insulin and those are gonna be the Max's but I just don't know which ones which he so unrefined that I need to plug back off function but a really high then or find out which is a matter of how because we studies level says these level sets positive and these levels as negative these definitely bigger than because of that Know these are positive and negative but will do it algebraically just to see that just follow the recipe For the death of 1 over Rupert to can't over route to remember at his exwife at it OK so this That is equal to onehalf And then have of 1 of route to negative 1 over to negative Taking those 2 . 3 That's also eagled positive 1 have been Of 104 route to negative on over 100 you aren't taking these 2 Here Volume and when I do that that equals negative So clearly this is the maximum this is minimum because this is bigger than the last 1 negative water ordered to positive water cuts also negative back so remember this process find you the locations of Max's by known advances in the is amended his did know which was which now I can say The use of the Mac's and his remit We conclude that after that while I was at exwife Subject to export plus guys where was 1 has made enacts about onehalf half the points Those points will hardly be derailed his a complete solution It does Millions of negative at those other 2 leaders those locations there we we knew in advance what they were when you're doing this Therefore these examples I do all the pictures and so it is From ask you to draw these pictures I wanna ask you drop threedimensional wanted to ask you to join us This is because business incorporates all the things that we've learned its that level sets it's got some some graph such circles on things and then seeing that there no gives you a picture of what's going on and they ask you Drive that picture There's more of media that I've drawn learn to draw the pictures along with the help Any question about the fact that the combination of dollar artist once we knew 1 and you're set The hardest thing about this is his deal with all plus or minus the track is what you are in the process Ole ole bachelor process that we didn't do together example is the only thing that changes in galleys pluses and minuses to keep track of 3 questions about that Look it's it's tile this together we're do couple but Yukon examples 2 There are too many scenarios when you talk about business he can't meet you have a business were the main thing danger interested maximum profit Maximum profits pretty important if no 1 minimum profit on maximum by minimum negative that's not good business so comes a calculus by in its own a business what are the things you're maximizing minimizing jab that street near mind so that you don't have to learn that as you're doing business Proud So OK we're Maximize profit maximize red knew of the money coming in for a battle that usually contributes to a maximum
39:13
Profit What's the other side of the if I wanna maximize the money coming in I wanna minimize the money going out Right so that minimizing the cost This cases with the budget for the least cost to the situation this get this manufacturer and they've got to products but in this this is the simplest case scenario we think of that accompanies keep in mind when we're doing this problem is that a company like General Electric religious got all these different products that they make that the narrator necessarily related like washing machines Dishwashers just 2 separate types of products so old Companies like that other companies like issued Mitsubishi General Electric user companies that make lots and lots different things so as soon as example examples to just think were really talking about these big companies that make lots of products do 1 with 2 examples but the real life stuff maybe has 50 different products You can still do all this with 50 products So that's why it's worth it to learn it for too because you can really extended any number of this method of Lagrange multipliers works for any number of variables Docket so here's the situation we have a manufacturer and a make to products Indeed the just have some example undermined you say this is HDTV using this as dishwashers The cost I know how much is If you can calculate how much it cost to make it can really make year and a half to get all the stuff ready to make these things you gonna know what the cost is You get out of formula they'll give you the formula the cost the cost of manufacture A certain number of these goods but say it would say we had actually bows and why it was so the cost is going to be a 6 x where players 12 Now you can actually see what this dysfunction Because we don't listen to seafarer move next that 6 of the 12 30 remove the 6 of 12 is just a parabola Drake so what happens when I modified It stretches out the wine 1 directional of more than EXE Nutri lips so this is an ellipsoid rather than a circular drive them it's a paraboloid but each 1 of these guys as ellipses each level said is similar to that of this guy looks like tender level sets early our lips and have a budget constraint X plus Y equals nite so execs plus Y X is a number of HDTV we create and uh Wise a number of dishwashers and what does this budget constraint represent it represents maybe that you can only shit 90 units a month normally that's you constraint MOI I'm only gonna be able to ship 90 so the question is do I make 30 of those 60 those er vise versa 245 of each 15 of those 75 of those would what's the optimal what's up what we talk about when it comes off up more we want the least cost so I find out the numbers that fits in here they gives me the least cost over here When you put that together that just means line of the gradients what the gradient is not the 1st with this level said to be Tianjin these little that happens at I'll be the 1st Well we just did the hardest 1 of this We've seen so much easier compared hookers couture process here we have different letters of of the gradient of S C representation at 1 solved that of doing gradient of making GENE put it in the context of what I really have very seen cast playing the role of the That function the previous examples land and that the constraints of this Trade here there should be a X Y equals X house why that constraints so we view it as a larger function and that's just 1 level said So in our new context we should have these symbols symbols but that process is still the same degree see that 12 eggs and 24 why gradient of B. that's just 1 and 1 ripped the derivative with respect Exs 1 derivative with respect to what I So I gave my 2 equations here I get 12 dead slammed 20 while he was land solidarity have Lander set up undiscovered take this just put it in there for Lambert is like than all the other examples I have 24 Why most well that's just like all the other examples at this stage I like in expect is to get the relationship between X and Y had to why that there is no relationship What we do is a step too We take that relationship with plug it into the constraint this case it's a budget but she replied X 2 X plus Y was 90 that's going be There's the actual values that Drexel so this stew lies in red and therefore X 2 Y Y equals 19 that implies posters concerns equals 30 than I can tell you that X is 2 times that 6 of the Magic point where these greens lineup is I manufacture 60 of TV's and 30 of the dishwashers but you see at this stage we still know that Imax mn clearly it's mn because
46:38
This which is being asked to do but Go through the process and check so I got the point at 16 30 vinify plugin here at about 60 30 tell you what that is plug in Can a large numbers there but I did at 32 thousand 400 unless check another point of check but put off another Point on the constraint we always have to use a constraint so what's easiest 180 0 9 year 45 45 45 5 . 45 is equal to 36 thousand or so this is larger than that 1 so just like in the previous example I knew I had a Mac sermon at this location said the plugin another point to see which 1 I had this case this isn't in that That configuration owner makes 60 HDTV 30 dishwashers and I will give you the minimum cost company Try So
48:23
I OK the most important application all these is that carbon Douglas
48:51
So if you see this in economics a Ivory covered in this class but if you seen in economics and seeing how Often shows a owed a big deal it is in summer the upper division classes So tours to practice this technique on this specific type of Docket so I let that happen example here Production according to labor and capital variables are laboring capital this is gonna be a 16 won 4th day of grief 3 common there's a couple of things here I've talked about this before The typical card Douglas Function it always has Fractional Expo images to give that curve That for years this way you what you wanna get that a lot diminishing returns for each Variable and this was maybe more extreme than that 1 our 1st They have dipped slightly different shapes but here's the thing Ms. is what make them out for fabric for Cobb Douglas Very often you have data Equals 1 And the reason I tell you that in advances that was gonna make Ollus algebra work at it may not be the case in this model that in real life PC but the matter is so much nicer when you make a palpable as equal 1 often examples do so when we receive direct so ever constraint however on the bottom or information here so L is number of units of labor and gave Number of units of capital And then tell you what each unit cost I get This is information you would have I know how much I'm paying my labor so Each community it This way Each labor unit costs 50 dollars each Capital unit costs A hundred dollars will talk about capital with thing about machines factories There various it should be more than label Often it takes a lot of the budget and then what constraint budget for for 500 thousand that so much we have Investing in this production We have a product that from the Hutto use this this budget year Well This is get equal 50 l plus 100 K race that if each unit costs 50 have 50 times Plus 100 times has Siegel that budget constraints of there's a there's er budget Mr. budget constraint is of fair play 50 L plus 100 K cost 500 So focus so of the user situation we've got but function which is more complicated than we had before better constraint which isn't too bad this is linear so you can marry CR derivatives again be pretty simple for this over there and it won't be so simple careful
53:27
I'm interested in order followed a recipe I want to
53:36
4 starts identify which is the function that scary key so what do of F Lambert times agreeing This is the constraint and this is the original function of this case now I have gradient he is equal to land at times But that's call this the budget right boats go town near the island He had to be careful about this when this was not a town but When I do the derivative with respect Elf non thinking of K is a constant summer bring them with me amending its 16 caters 3 quarters as I change loses constants nominating a derivative with respect to L and somber multiply in front by a quarter and then I'll have L missiles
54:47
The the images right outside committee began moving it for years
54:54
As multiply front of the 1 for them and get down to the negative 3 forests
55:01
And then I came to the fore to see you believe that announced to the derivative with respect to Cary I'm going to keep their L 2 onefourth That be constant multiply in front by threefourths of the 16 member force for the Times 3 12 year and the cave stays
55:33
The case is what I'm doing the derivative so I have 3 quarters minus 1 located and negative way cause It's not quite revealed At this stage wired so nice that have those
55:50
The exponents add up to 1 that is least you can see here because they added up to 1 c ended up with the same numbers here because they added up to 1 he got that so that's why that happens in the pocket got my green P animal that that equal to lambda times the gradient of but that is easy to compute the grade B B C L
56:19
That's 50 and he said OK that's won't tolerate that over the beef that is equal to these K equals 100 so that's the beauty of the week OK so you've got to equation is not that nice Redden down 4 L negative threefourths to the 3 quarters equals Lander And then I got 12 L little onefourth of a negative 1 4 equals 100 but this point you may be tempted to use give up his You know maybe Just don't his losing your concentration it's hard to keep track of it at United ensure you doing it right at this point is you go these exponents that you're not familiar with the Yankees keep forging forward so what we do in every single case we sulfur Lander And took whatever was plug didn't remember a year and then we saw we got desolate do that here slammed a 5th off yet land that is for L and negative 3 force the 3 wars that this part and I had to do that I've a 50 foot tall and now this is equal lambda toroid do land I know I can't simple fighters to make it to over 20 and that's fine you could do that if you this was quoted in the there But go here that almost there you'll be surprised Canary see Summary reproduce that last line I have 12 worth a 4th Wills 100 Times slammed the mountain and replace slander With that big monstrosity you're there for L the threefourths of a negative forces over On the 1 source that this Mrs. Give me 8 each of the 3 worst case negative And this is where the matter can be saved if I just multiplied I wanna get all the cases on 1 side and nowhere else on the other so if I feel I have something He thinks he is Negative takes a Getty get that right OK Let's multiply this by over the 3 forces Bell the 3 forces of the poverty and multiply this side by case is the onefourth that vacated the onefourth And Others right but I just said I'm multiplying both sides by that helped the threefourths of a point to hear the case canceled over here they now Els See multiplying those sides but that now what I have is 12 El Pais that Relationship equal to 2 OK is equal if she has whatever you like We due This was the 2nd step here I know this is the 1st that we got a relationship between the 2 variables now in the 2nd step we take that relationship plug it into the constraint and give you the actual a part about this all the Expos but why not practicing here means that if Added any Yukon glasses are ready and mastered it well and you're OK with this this is kind of an easy example but if you've never seen it before the new practice here than take it any kind class you will obscene it wants an easier the 2nd step to honor I have L plus 100 K equals 500 thousand But I just found out this relationship solidstate kid equals that Put here Have 50 L plus 100 times 3 have equals 500 thousand the 200 close above 100 thousand 100 and K is equal to Adds times 20 500 36 50 and this is this is your Mac's production on this to you we could to check this Another value She she If the farm a high the constraint We had a very nice of location as the values This is the number of Units of labor that and use the number of units of capital for the Mac's I didn't bother to plug that enter the calculator
1:03:32
I don't on a couple of days do to plug defined P is there a way that's the maximum production off the key thing there is
1:03:45
Practice with those that because these are the functions this
1:03:48
This is what models society so that you can avoid fear an economist you look under the real world and you see what's going on
1:03:57
The math is forced on you Can't decide your function of these the functions that described the real world suffered in the study the real world you have to do that Questions about that these steps they're all pretty clear Key In switch gears start reviewing them on 1 thing that
1:04:33
I didn't Finished doing was valued at least So we spend all our time lag value theory we did examples of 2 by 2 matrices but what's left over the 3 by 3 matrices so by way of review and an addendum on their remind you about the agonizing stuff in the 283 by
1:04:53
3 matrix they you have that to practice over the weekend and on Monday when I give you the sample final problems still be another 3 by 3 may I promise I'll be at 3 3 matrix on the final so you get to practice 1 do today and you get a new 1 on Monday I'll do that for you help you out with that and that the those 2 examples of maybe practice more He will go over that stuff they question about this double a race that All along There But looking at all There's a matrix 3 by 3 and the command here it defined the a I values And eigenvectors of and then construct you That blood Diagonalize is the 3rd thing is to work You glamor which is betting matrix focus who we are As we do this example I'll remind you of the recipe they you follow to get all these things 2 For look at the 1st thing you do you find the idea values in the way you do that is you solve this equation Well you can't just start solving it at 1st dissect with the notation vertical lines here that means take the determinant in a minor slammed I'll really all were doing there is were subtracting landed down the main backer of this guy here a minor slammed I that's equal to This matrix oneliners and 0 1 0 1 9 slander 1 1 1 2 That's the way minor slammed and then the determinants of a minor slammed That could determine the list Andrew Wednesday with 0
1:08:45
Tonight to take the determinant of this 3 back 3 matrix lets us do the determinant along this column
1:09:02
A side we do determine its this guy times the determinant The resulting the leftover matrix after across offered to role its column there'll be this is plusminus less than a B minus 0 times the Matrix leftover plus 1 times nature of Dover so they read all that at this is this gives us this equation have 1 minor slammed kinds of determinant of the Matrix that's left over when across those off so that 1 minus slammed 1 1 2 Yes Well I should've area Thanks to the that's too late to save focus so that's the first one that I have 0 times the leftover that doesn't count for anything in the summer 1 times the determinant of the Matrix leftover 0 1 1 minus slender 1 seat you leave me alone And were setting the sequel 0 solicitor equation we had him quite that equation yet we're were just unraveling at this OK so the determinant here And that 1 minus landed times a determinant here which is 1 190 slammed a time to minus slammed a minus 1 of just parade here is the determinant of that nature and then I have minus 1 minus 1 time 0 at the city and its positive 110 0 minus 1 minus and that's all I have 1 time 0 minus 1 . 1 0 that And this is all equal Gaza if
1:11:42
A dual algebra trickier here At this stage we do Could you could you could just start multiplying all let out and get upon only on A factor to solve it BC already got somethin factors almost factors noticed there's this whole term here and there's another term here and they both have a Islam So in since I'm going to be factory starts factoring now I've got this The factor that up on a red them but it's can algebra trick that you might not think of unless somebody shows it to So it is reproduced them over there have when 9 slander parentheses in that would minor slammed a few minor slammed a time of 1 minute minus 1
1:12:47
Equal 0 votes should be the same as that yes sir So what I just said Wasserman a notice that the I Have a oneman Islam each those factor that 1 minor slander and then OK If I factored out 1 minus slammed from this and this whole thing left that 1 minor slammed a minus Lambda minus 1 and then a factor that out of another minus 1 This part is issued a 1st term and that extra minus 1 4 minutes but will take PCs Taiwanese this trick as it makes it little OK Mallett multiply out on inside here Where multiply that I get to minus 2 lambda minus slammed and then plus Landis squared minus 2 that's what I did that because look to cancel I have 1 minor slammed and this is all equal I have 1 minor slander now what I have on inside here I just said Landis squared minus 3 land and now I'm home Greek 2nd factor that into land times minus 1 cursor 1 minus when the church from factoring out of the land minus 3 left now I have my Landers and 0 Schools On your 3 But algebra but that's a lot algebra The Lecont our ideas that it's not for each item value provision eigenvectors It's nice about this was will get a chance Practices 3 times nonetheless finds nagging sectors That's the 2nd step of the way I do that is that for each Lander found in Part 1 would solve the equation a minus some land lie here we don't have determined were just doing this this matrix times that equals 0 and Now I have 3 of a market column each time because I wanna keep track of salt water will be you want will more than 81 of will double but in each case and solving for the vector here and he got 3 component I'm expecting to get an answer with 3 values Are you can That OK so let's go let's do our 1st So 1 land equals 0 solve that equation that employees land equals 0 and she considers inundated be land the is 0 times I was 0 everywhere illustrate that at a minor slammed that's just a again as I'm innocent drag 0 from each of the decade also just before major cracked again
1:17:12
And then lets a minor slammed by you That's can it be that equal 0 that I had this equation 1 0 1 0 1 1 2 times you 1 2 3 of That's our system of equations matter solve the system of equations I put in an augmented matrix 1 0 1 0 1 1 1 2 0 0 0 So if you we try to reduce the end and if I'm putting in a row Echelon form the 1st thing I see is I'll take this 1 here in the upper lefthand corner nominee use it to you that once I get so I'm gonna do here I'll do a negative wrote 1 With see a minaret Growth rate of negative role 1 class of the role of zeros never changes caused no matter what I multiply this by an added never changes anything she does reproduced that the top arose not gonna change the 2nd rose not Onitsha that not doing anything to was the 3rd row will change And what's again and be the first one near 0 it's negative 1 plus 1 0 then 0 plus 1 1 and then negative 1 plus 2 His goal for I don't OK On the that I could remove the last row I wanted to have to do is replace this with negative too Plus row 3 D that I replace wrote 3 by negative growth to last 3 9 1 0 1 0 0 1 1 0 0 0 and that's in reduced I can't I can't charity if I Cheney readers won the nomination ruin that 0 so that's the lowest form write down what we have this guy says that would how you would you to bless youth 3 equals 0 and that tells me that I knew too is equal to her that's negative 3 And this here says you want plus you 3 equal 0 that implies that you Want So what we learn what we learned about the veteran interested an interesting in which is you wonder Due usury 3 I just found out that you want his evil the negative view 3 was put that I just found out that you too is equal to make it if 3 And you 3 that's equally you 3 So this is equal to use 3 times negative 1 negative 1 Wanda not you like you can set you treat equal the teeth you want to say 3 feet then you is equal to negative negative T CD native when they want to stress savior step value need is a relationship between these things and the new factor that out the the result the leftover matrix that Cher eigenvectors off talk XOU only I want to go faster because it's the same process any questions about that closes the chance to ask about that
1:22:13
Give us a jerk So I got year If you think Our By the determinant yet There are battling to 2 1 0 0 1 0 But well 1 Back Mexico's higher Yet again in nature of Lambda Phi that times 1 0 0 0 1 0 0 0 1 of the New landed each 1 of those So when subtract urges attracting over there A request and talk but goes through this path of ballots to but their their land and land equals 1 So here were gonna do do a minus 1 I kind equals 0 Walsall that a minus 1 I ileitis means subtract 1 from the bag this is going a minute Attract 1 farmer that no I get 0 0 0 1 0 0 1 1 1 1 to believe Yes year Yes They turn You want 1 or more and and I'm aware that this is just how I did it so that My vector for you might get 1 1 negative 1 of you may get to 2 negative to you when you pick it up when you do your way you make it different ones But I'm aware of that so I check for them agrees OK so I'm looking for Didier so that anyone lead to be greedy They 0 0 0 and then we can translate that system of equations into an augmented matrix had 0 0 1 0 0 1 1 1 1 start reducing C each year I actually don't even need to reduce I I just cross that off as I'm gonna get rid of it right across the 2nd not their way that's fine lips The bright red Down so I minaret place 1 With Negative Road Chiu plus 4 0 1 and the matrix I get is 0 0 0 0 0 1 and here we actually get some information here this 1 says that he is 3 0 and this 1 says he wants plus 2 plus 3 equals 0 but that goes there I get in 1 place 2 0 ran space on School over here That says that as to leave negative 1 so would I find out about the vector as anxious to be his 1 to 3 kV 1 was just got 1 and then to his negative 1 Jews negatively 1 3 0 1 that's like it Now like you can't you can't do this again over just say 2 equals tea or something that's an extra step that 1 hand I'll be makes it easier for the ideas you just shiny relationship between all the vectors and then you wanna have the same variable in here just factored out whatever's left over Israeli
1:27:20
Accrued Well we're almost there is no doubt 1 or more of the same process
1:27:33
Hopefully there'd be a little more That is the case of landing equals 3 and now didn't do a minor slammed I sobbing on 1 minus 3 period negative to 1 minus 3 period negative to to minus 3 You're a native might a minus 3 5 w 0 when I do this I get negative choose 0 1 0 NEGATIVE June 1 1 1 negative get Times W 1 W. W 3 0 0 0 same thing is to go out extra to Here's but that in the augmented matrix 2 0 1 0 1 1 George we do far valid state this 1 right here An era that negative to thought to replace Wrote 1 with that too Roast 3 plus 1 just wanna make some more zeros not solve of row 3 is not going to change route to snuff exchange But when I do positive 2 times this season 0 Positive to add it to positive to negative to an added
1:29:18
It's aimed at So let's say that little more more care
1:29:34
I'm getting get rid of this role here Let's get become 0 0 0 Adelaide you that disconnect replace row 1 death road to plus 401 salutes we have OK so this here that says what negative Q W Q Plus W 3 0 so that the W 3 to W 2 In the last year over here that says W. bond plus W to minus W 3 0 0 but we know the W 3 Calls to yes I urged that the moon is His negative is when across it added disappeared that was I've got this relationship W 3 equals W to come put that in here there and I have W wannabe plus W to minus 2 W 2 0 and that's as the W 1 calls W2 Go there reproduce I just said its kind all over the place But So we get a W 3 quarters to Dublin to and then I got a W 1 most of these series things in terms of W2 toward a learn about the vector interest and W 1 W. jumped up 3 Well W 1 that WTO W 2 is also W2 and W 3 2 measure at the after practices for a little bit more visits
1:32:21
It didn't reduce down like the other ones OK so now we've got everything we landing equal 0 We got the vector world leaders you would want 1 negative or of different than what I want is a classic valid different I notes here and then for land equals want be is that 1 over here still
1:32:59
Where is the 1 they want 0 and that plant at Serie a that sector doubly all the workers
1:33:18
Done more most of the work is done in finding this vectors want you have the vector yes but NASCAR It's the same things as multiplied Is that reality is the 1 that I had on the board for that I did have a
1:33:37
6 it has ever have OK if so that the last step is not up there anymore because on the directions of their but it says Find Matrix the dagger Liza's any of that nature is Q and the way we form Q is we take these I again values and put them in the car park eigenvectors and put the columns for the church but the 1 thing we have to do we have to divide by the likes of these vectors here have to beat length 1 Q. has to be an orthogonal nature But I remember what I meant by Aristotle matrix anymore you should look that up for the final but An orthogonal matrix every column has linked 1 and they're all perpendicular notice that these are all perpendicular if you do that Inner product of all those all perpendicular OK this is how you create that nature matrix this committee each 1 of these guys and abide by the lengths So the length of you that's equal to rue 3 and the length of that's equal to repetitive and length of W that's equal to 1 squared plus 1 squared is too close to square for plus 2 6 Those are the lengths and now I can ride at the final few here and give yourself some space focus on taking you an undivided by 1 over Ruth negative 1 over Route 3 negative water over Peru 3 1 3
1:35:29
And then he and dividing by Beirut to over to negative want to 0 In the last 1 . 6
1:35:42
What group 6 2 so that is the matrix that Dagmar is
1:35:50
A which means Q injures a Q that Beagle 2 lander Which is a diagonal matrix And we can raise without doing their multiplication we can just say what land because we know in advance that of the economic use it you get Ask that the values and they have to be in the same order so this wondrous remained equals 0 this is planned equals 1 missiles land equals 3 so I have to have that order 0 1 3 and that's the solution to that problem To save yourself a lot of time if you if you are willing to take maybe 45 minutes a half hour depends on your level but has Treasury produce this if you can get that done over the weekend and on Monday when you watch me do another 1 yielded Hilda seems so easy could you will have gone and done this work and LCD allowed a timely and ready for the final yet Look up Show that Process of diagonalization is I create this matrix and then I do this multiplication if I were to do the multiplication I would get this nature but the process of diagonals Asian is a user's Matrix multiply neither Saturday in burst and the regular matrix and you get the adding value so we can know that advance so so I'll have to do the multiplication is right down here is the agony value that goes with this I did in the upper lefthand corner Thursday in value for that in the middle so you had put this sector hearings which the Sudanese leader which that's why I put the values They go with the vectors So we know what the stag no matrix is going to be at a time in fact I really knew what it was gonna be after found the ideal Judd Rose There a diagonal nature so diagonal matrix the only place it has nonzero things is on the there in house an essentially what you've done is you you've you've extracted the essence of this a matrix the real information is that these these adding values that's how much it stretches things in each direction
1:38:27
OK so a lot of practice but you you can learn a lot but is taking the same problem is doing it over and over again because really it's this algebra here they need practice concepts launching memorize that that's not hard algebras which in practice
1:38:43
Just keep doing the same want to get really good at 2 Do Yes knew land because the determinant of a is equal to the product of the and Tracy political Said don't have it does apply has 30 Croatian busiest 3 variables so it's going to be a landowner 1 landed to land 3 because they determine today The sum of the still be the trace but I need a 3rd equation and that's beyond the scope of this class to get rid of its theory is that An analogous version of that for 3 by 3 you napkin abilities And it's not it's not any easier Show them Now there's still up there you feel freer asked but that I think Judge What that off the algebras Messier that Because have 3 variables for every 1 of those systems that the only thing that's different well and then the determinants Messier but gets the same method that things and that things differently trooper 4 by 4
00:00
Resultante
Offene Menge
Nebenbedingung
Prozess <Physik>
Kalkül
Punkt
Extrempunkt
Klasse <Mathematik>
Rechteck
Zahlenbereich
Kartesische Koordinaten
LieGruppe
Überlagerung <Mathematik>
Übergang
Richtung
Eins
Gradient
Ausdruck <Logik>
Multiplikation
Variable
Exakter Test
Stichprobenumfang
Vorlesung/Konferenz
Zusammenhängender Graph
Tangente <Mathematik>
Gerade
Analysis
Lineares Funktional
Mathematik
Graph
Mathematisierung
Vektorraum
Biprodukt
Fokalpunkt
Kritischer Punkt
Menge
Ordnung <Mathematik>
08:05
Konzentrizität
Nebenbedingung
Lineares Funktional
Kalkül
Punkt
Menge
Reelle Zahl
Vorlesung/Konferenz
Übergang
09:37
Radius
Nebenbedingung
Lineares Funktional
Punkt
Sterbeziffer
Kurve
Gleichungssystem
Ungerichteter Graph
Physikalische Theorie
Gradient
Übergang
Differenzkern
Rechter Winkel
Vorlesung/Konferenz
Ordnung <Mathematik>
12:13
Quadratzahl
Vorlesung/Konferenz
Gleichungssystem
Derivation <Algebra>
Ordnung <Mathematik>
Gradient
14:10
Resultante
Nebenbedingung
Punkt
Gewichtete Summe
Graph
Extrempunkt
Fortsetzung <Mathematik>
Gleichungssystem
Fokalpunkt
Eins
Übergang
Modallogik
Strahlensätze
Weg <Topologie>
Negative Zahl
Quadratzahl
Differenzkern
Menge
Flächeninhalt
Tourenplanung
Mereologie
Vorlesung/Konferenz
Analysis
19:43
Vorlesung/Konferenz
21:01
Nebenbedingung
Punkt
Zahlenbereich
Gleichungssystem
Kartesische Koordinaten
Derivation <Algebra>
Ungerichteter Graph
Term
Übergang
Eins
Gradient
Negative Zahl
Variable
Einheitskreis
Inverser Limes
Vorlesung/Konferenz
Zusammenhängender Graph
Drei
Lineares Funktional
Kreisfläche
Graph
Physikalischer Effekt
Frequenz
Ereignishorizont
Quadratzahl
Menge
Verbandstheorie
Forcing
Mereologie
Kantenfärbung
31:02
Beobachtungsstudie
Nebenbedingung
Lineares Funktional
Kreisfläche
Punkt
Prozess <Physik>
Ortsoperator
Mathematik
Graph
Extrempunkt
Physikalischer Effekt
Wasserdampftafel
Kombinator
Fortsetzung <Mathematik>
Fastring
Eins
Gradient
Übergang
Weg <Topologie>
Negative Zahl
Differenzkern
Parkettierung
Tourenplanung
Vorlesung/Konferenz
Spezifisches Volumen
Schnitt <Graphentheorie>
39:09
Nebenbedingung
Trägheitsmoment
Subtraktion
Punkt
Prozess <Physik>
Zahlenbereich
Gleichungssystem
Gradient
Übergang
Richtung
Ausdruck <Logik>
Gruppendarstellung
Erwartungswert
Variable
Multiplikation
Einheit <Mathematik>
Reelle Zahl
Vorlesung/Konferenz
Gerade
Trennungsaxiom
Lineares Funktional
Erweiterung
Oval
GreenFunktion
Güte der Anpassung
Biprodukt
Auswahlverfahren
Ellipse
Parabel <Mathematik>
LipschitzBedingung
46:27
Nebenbedingung
Prozess <Physik>
Punkt
Extrempunkt
Zahlenbereich
Vorlesung/Konferenz
Kartesische Koordinaten
Konfigurationsraum
48:47
Umwandlungsenthalpie
Lineares Funktional
Nebenbedingung
Bruchrechnung
Kurve
Klasse <Mathematik>
Zahlenbereich
Derivation <Algebra>
Biprodukt
Division
Arbeit <Physik>
Variable
Einheit <Mathematik>
Vier
Einheit <Mathematik>
Reelle Zahl
Minimum
Vorlesung/Konferenz
Faktor <Algebra>
Ordnung <Mathematik>
Numerisches Modell
53:34
Konstante
Nebenbedingung
Lineares Funktional
Multiplikation
Rechter Winkel
Vorlesung/Konferenz
Derivation <Algebra>
Gradient
54:53
Nebenbedingung
Punkt
Wald <Graphentheorie>
Exponent
Physikalischer Effekt
GreenFunktion
Klasse <Mathematik>
Abgeschlossene Menge
Zahlenbereich
Gleichungssystem
Derivation <Algebra>
Rechnen
Biprodukt
Gerichteter Graph
Gradient
Konzentrizität
Arbeit <Physik>
Weg <Topologie>
Negative Zahl
Variable
Einheit <Mathematik>
Differenzkern
Forcing
Mereologie
Vorlesung/Konferenz
Gerade
1:03:29
Beobachtungsstudie
Lineares Funktional
Matrizenring
Mathematik
Extrempunkt
Vorlesung/Konferenz
Biprodukt
Physikalische Theorie
Numerisches Modell
1:04:46
Matrizenrechnung
Zahlensystem
Knotenmenge
Determinante
Stichprobenumfang
Statistische Analyse
Vorlesung/Konferenz
Gleichungssystem
Fokalpunkt
Diagonale <Geometrie>
Gerade
1:08:49
Matrizenrechnung
Flächeninhalt
Determinante
Natürliche Zahl
Vorlesung/Konferenz
Faktor <Algebra>
Fortsetzung <Mathematik>
Gleichungssystem
Zählen
Term
Fokalpunkt
Teilbarkeit
1:12:35
Matrizenrechnung
Abstimmung <Frequenz>
Wasserdampftafel
Gleichungssystem
Vektorraum
Teilbarkeit
Weg <Topologie>
Polarkoordinaten
Eigenwert
Mereologie
Vorlesung/Konferenz
Zusammenhängender Graph
Luftreibung
LieGruppe
1:17:02
Resultante
Matrizenrechnung
Subtraktion
Prozess <Physik>
Determinante
Mathematik
Sterbeziffer
Natürliche Zahl
Klasse <Mathematik>
Gleichungssystem
Vektorraum
Physikalisches System
Bilinearform
RaumZeit
Teilbarkeit
Eins
Negative Zahl
Eigenwert
Vorlesung/Konferenz
Normalspannung
LipschitzBedingung
Nominalskaliertes Merkmal
1:27:16
Matrizenrechnung
Negative Zahl
Prozess <Physik>
Ortsoperator
Rechter Winkel
Tourenplanung
Vorlesung/Konferenz
Frequenz
Aggregatzustand
1:29:18
Negative Zahl
Reihe
Vorlesung/Konferenz
Vektorraum
Term
Einflussgröße
1:32:19
Reihe
Vorlesung/Konferenz
Vektorraum
Eins
1:33:36
Matrizenrechnung
Negative Zahl
Länge
Eigenwert
Wasserdampftafel
Natürliche Zahl
Vorlesung/Konferenz
Vektorraum
Fokalpunkt
Skalarproduktraum
RaumZeit
Richtung
1:35:16
Matrizenrechnung
Prozess <Physik>
Natürliche Zahl
Besprechung/Interview
Gruppenkeim
Ideal <Mathematik>
Vektorraum
Richtung
Übergang
Diagonalform
Multiplikation
Vorlesung/Konferenz
Ordnung <Mathematik>
Diagonale <Geometrie>
1:38:15
Variable
Gewichtete Summe
Determinante
Klasse <Mathematik>
Vorlesung/Konferenz
Gleichungssystem
Physikalisches System
Biprodukt
Drei
Physikalische Theorie
Metadaten
Formale Metadaten
Titel  Math for Economists  Lecture 14 
Serientitel  Math for Economists 
Teil  14 
Anzahl der Teile  15 
Autor 
Kronewetter, Jason

Lizenz 
CCNamensnennung  Weitergabe unter gleichen Bedingungen 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nichtkommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben. 
DOI  10.5446/12908 
Herausgeber  University of California Irvine (UCI) 
Erscheinungsjahr  2013 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 