This channel contains the video lectures for the Economics Department's Math Camp at the University of Arizona. Arizona Math Camp prepares students to enter the Department's PhD program.
To see these videos organized in the intended order, in playlists organized by content, scroll down a bit in the Home tab.
There are lecture notes and exercises for the course here: www.u.arizona.edu/~mwalker/MathCamp2020.htm
These videos were a collaboration between my TA Risheng Xu and myself. Risheng operated the camera for all the videos created prior to 2020, and more important, he solved a number of technical problems: how to get high quality when putting LaTeX text into Lightboard videos; how to put high-quality graphics into Lightboard videos; he helped me solve some lighting problems; and many other issues that had to be solved. I'm enormously grateful to Risheng --- and you should be too, if you've found these lectures to be helpful.
Thank you! I found these courses are extremely helpful. But the lecture notes seem no longer available(I found the link on your home page but the website which it direct to is not found, they might have been archived.) I will be more than appreciated if you could kindly provide a new link. Thank you!
It's that time of the year agat. This is the best lecture for Econ math camp! Exactly the pace and depth we needed. The link to the lecture notes isn't working though, any way to get the lecture notes?
As I mentioned right at the beginning, it's in an earlier lecture. Note that lectures 36(A&B) are titled "Solution Function and Value Function." It's also identified in this lecture (38A) at the 7:38 mark and subsequently.
For the pre-image of the origin, (0,0): Ax = (0,0). So 2x_1 - x_2 = 0 and also -2x_1 + x_2 = 0. Both equations yield x_2 = 2x_1. For the pre-image of the point (1,-1): Ax = (1,-1). So 2x_1 - x_2 = 1 and -2x_1 + x_2 = -1. Both equations yield x_2 = 2x_1 - 1.
This is amazing. Vert nice step-by-step introduction to KKT conditions which elsewhere including lectures by Boyd were very confusing for me are super well explained here. Especially this video shows the non-triviality of the whole issue (something I totally missed elsewhere). Thank you. You did a great thing for humanity by publishing this online and free to watch.
I'm glad it was so helpful, and thanks for the great comment. (Maybe I wouldn't go quite as far as "a great thing for humanity", but I really appreciate the accolade.)
Hi! I’ve just finished the 105 videos playlist and I’d like to sincerely thank you for putting those online. You’re helping thousands of Econ students worldwide (and attracting a lot of attention to the University of Arizona) - greetings from Brazil.
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About notation, early in my first semester my, advisor showed me a paper he was working on. I was stumped. I couldn't figure out what he was doing with the matrix derivatives, and became very anxious. After a few daya, i told him that i couldn't figure out what was going on. He was like what matrix caclulus, so i showed him he had matricies with primes all over the place, and he's like it's the transpose. I had only ever seen the transpose denoted with a T.
Yes, notation can be tricky. There are lots of things for which different people use different notation. Matrix transposes is just one of them. There are a number of places in my lectures here where I indicate that some people use notation that's different from the notation I'm using.
I don't see how this is unique to lexicographic preferences. Consider any strongly monotonic, strictly convex, and locally non-satiated preferences in R^2. You can draw a similar image as you have done such that u(a) < u(b) < u(a') < u(b'). And the rest follows as before. But this would imply that the familiar diminishing MRS preferences don't have a utility representation.
The proof requires that we can do this for *every* x' that's larger than x -- i.e., if x'>x then it has to be the case that (x',0) is preferred to (x,1). That can't happen for every x'>x if the preference is continuous: continuity says the weak upper-contour set of (x,1) is closed, so if a sequence of points (x(n),0) converges to (x,0) and every point in the sequence is even weakly preferred to (x,1), then (x,0) must be as well -- but we know that in fact (x,1) is strictly preferred to (x,0). And of course the representation theorem tells us that if the preference is continuous then there *will* be a utility function for it, and in that case we can't do the construction of the function f(.) in the proof.
If this was a minimisation problem, would our derivatives of the function be ≥ sum of the linear combination of the derivatives of the constraints (line 2 of KKT Conditions) (rather than ≤) Also why do so many texts on KKT miss out this condition of ≤ and just set the derivative functions = to each other, like in an equality constrained problem? Do you have any reliable texts you recommend? Thx if you get time!
That's the right idea, but it's not quite as simple as that. An important key to dealing with minimization and with >= constraints is to note that (a) minimizing a function f is identical to maximizing -f, and (b) multiplying both sides of a >= constraint by -1 turns it into a <= constraint. This will enable you to apply the KT Conditions as given, or alternatively to convert them to conditions for minimization and/or >= inequalities. (How has your year gone so far?)
@@ArizonaMathCamp Hey, good to hear from you! Apologies for my delay I am in the peak final exam season for mine, six-year pilgrimage through Econ/Maths. Frustratingly, the KKT conditions are still haunting me. The optimisation course I’m currently using for example does not even note the condition of our Lagrange derivative j being ≤ 0 if xj = 0. It also handles minimisation problems by just subtracting the sum of constraints…I.e. it doesn't multiply both sides by, -1. I just find the KKT conditions are taught in so many different ways!!
The textbook for this current course is built off is - Sundaram, L.R. A First Course in Optimisation Theory. I don't find it that great at times. Do you have a single source of truth for Lagrangians the KKT conditions? With good practice questions for all the edge cases? (obviously your material is great, but doesn't cover constraint qualifications for example).
@@cormackjackson9442 Well, you've already cleared the first hurdle, by recognizing that there is not just one CQ. A CQ is a condition *sufficient* to ensure that the first-order (necessary) conditions will hold at a solution (a maximum). A number of different CQs have been identified, each one sufficient for this. Most treatments give just one CQ, which obscures this multiplicity. I like Sundaram's book a lot, but it has just one, I believe. The only book I can recall (it's not the only one there is, I just don't recall any of the others) that covers multiple CQs is a book by Peter B. Morgan titled "An Explanation of Constrained Optimization." The book provides good intuition, and discusses multiple CQs, with examples.
Writing on a glass screen, in the normal way, with camera on the other side, and then flipping the resulting video via software. Watch this: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Adm2raocQZ4.html Matt Anderson uses a mirror in order to do a live presentation, but I flip it in post-processing. If an ad comes up at the beginning, click "Skip" in the lower right corner.
@@ArizonaMathCamp Thanks for the explanation. More importantly, thank you for the content. It assisted me with passing my discrete mathematics course. It is much appreciated.
Hello! I'm sorry that it's an off topic question but, would your courses be suitable for a first year computer science student? I have my finals in 2 weeks and your teaching style is so engaging! I hope you're doing good and thank you for making these public!
Thanks for the kind words. I can't really give you an answer to your question -- I don't know what math your computer science course includes. Good luck on your exams!
Yes, that's correct. The antecedent in each definition is: for every set V [in the target space] such that f(x-bar) is "in" V [i.e., an element of V for a function; a subset of V for a correspondence]. And the consequent in each case is: there exists a set U [in the domain] ... etc.
Professor...I have a question here. If f(x) is a subset of V then the intersection between f(x) and V is not a null set. This is of course not true the other way as you showed. But if we replace the part f(x) is a subset of V in the definition of UHC by this implication then I am arriving at the definition of LHC which doesn't seem right. I am doing something wrong here for sure. Will be grateful if you help me to clear my doubt
I'm not clear what you mean when you say you're going to "replace the part f(x) is a subset of V in the definition of UHC by this implication." In particular, I don't see what you mean by replacing the subset statement with an implication.
People like him are the best people on Earth and make me not lose hope on humans. They're a big example of education doesn't have to be profit oriented.
That way is OK. The key fact is that if λ is 0 or 1 then the two sides of the inequality are the same, so they're equal. So you do have to use λ ∈ (0,1) for strict concavity; but it doesn't matter which you use for concavity.
Would we not be able to prove that the empty set is a closed set using the same kind of vacuous proof and thus contradict ourselves? And couldn't we make many nonsensical arguments using this technique. For example, if the argument in the video is okay, then what's wrong with arguing that for every x in the set of natural numbers, x in the empty set implies there exists a y in the empty set equal to x + 1, showing that it's vacuously true that the empty set is an infinite set?
You're a little bit confused about open and closed sets. See my Lecture 10. The empty set *is* closed as well as open -- that is not a contradiction. Its complement (the entire set of which the empty set is a subset) is also both open and closed. In other (non-Euclidean) metrics, other sets as well can be both open and closed. In your second example, you've assumed that there is an element in the empty set (which is false), from which you derive that the empty set is infinite, which is also false, thereby showing that the original assumption must have been false -- indeed, a proof by contradiction, or indirect proof.
@@ArizonaMathCamp Thank you! I think I'll need to carefully work through some examples to clear up my confusion. Thanks for the lectures, they're very helpful!
Yes, carefully working through examples is absolutely the best way to gain a solid understanding -- especially if you have someone who can tell you whether or not you're on the right track. Note, by the way, that it's important to recognize that open and closed subsets don't work the same as open and closed doors, windows, etc, where it's binary: a door is either open or closed and never both, but a subset can be neither open nor closed, or both open and closed.
From last few days I was going through a lot of books and topics to get a clear cut idea about this theorem but everything were being in vain...but after watching ur represtation on this topic i dont think i need to see anything else...great teaching prof...thank you so much
Professor please help me: I was told a Euclidean space is just a affine space with an inner product space. Does an inner product space automatically mean the Euclidean space is also a norm/metric space? If it doesn’t automatically assume this, what exactly can we do on the space if it’s just an inner product space (without being normed/metricized)? Finally - if we do assume inner product means normed and metricized space, does this then mean we can add/multiply vectors, measure lengths, measure distance between points, add a point to a vector (although not add point and another point - which I geuss doesn’t even make sense in a vector space anyway let alone affine space). Thanks!!!!
Hey would you answer a question for me: what’s the difference between E^n and R^n? I thought a Euclideaan space is E^n! Help me professor! Also what is the difference between a “Euclidean point space” and “Euclidean space”?! Thanks!!!
Super clarifying class. I like the way of teaching of professor❤. I am going to participate in a competitive exam and scared little bit, so please tell me how can i clear that exam? I am waiting for your honest answer😊❤.
Thanks for the videos. A nice and clear refresher on the topic! However, I believe that the explanation for the case a=0 is not correct. Even if a=0, Q(x) = xAx is still a quadratic from and hence Q(x)=Q(-x).
Thanks for the lecture, Prof! Although I know that the symmetric and completeness properties are different, the notations you used on the left side show they are equal.
You're referring to items (b) and (d) on the left. Note that (b) is an implication and (d) is a disjunction ("or"); they are not the same. Here are two examples of relations that are symmetric but not complete: x is a sibling of x', among a set of people; and the "unequal to" relation among real numbers. And an example of a relation that's complete but not symmetric: the weak inequality relation among real numbers.
Oh Gosh, I can't believe that I come across this video created by the inventor of the ->-> notation! And I can't believe it has only a few thousand of views so far, making me feel like unearthing a treasure trove. Thank you so much for your lucid explanation to these exotic concepts, way better than my professor does.
