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... hopefully with not too many bugs: here. 

Update: now with hopefully fewer errors.

I'm in office today, and have talked to four of you ... and got two errors pointed out. In case you intend to bring those notes for the exam, you might want to update. 

Recent updates (i.e. May, two of these updates have been announced here): 

• Today: the convexity/concavity lecture note, page 14. Drawing was correct.
• Today: the quasiconcavity lecture note, page 6. Formulae were correct.
  • I also put a note down the lower-right corner, but that is not a correction. 
• May 15: The soluti...

2017 and 2018 solutions, page 4 in both. As always, I am as grateful over the corrections as annoyed over the need for them, please bring them on.

The request I got for Thursday was about what I intended to start at. 

I didn't specify for today whether room 1220 (and when?) or not -  but just knock on my office door if you have questions.

As for Thursday: though I've had a few questions on my door, I can't remember any requests - so if there are any, please bring them on. We could take it there-and-then as well. 

I will in any case start out with last year's exam. Likely with the most horrendeously answered problem. 

16-18 again tomorrow.

(There's a theory reading group which I finally have the chance to attend 12-13 - and furthermore it wants the room.)

I gave a choice here. Sondre suggested two full exam sets. If you have other priorities, then e-mail him with copy to me, and I will try to produce a solution set quickly; otherwise, I'd be more than happy to cover the leftovers on May 16th. 

Nils

... exam 2015.

And either one more complete set, or some "bits and pieces".
Edit: do you have preferences?

p8, two "1" corrected to "2".

• As those present at the workshop know, I had forgotten the discount factor in the dynamic program problem statement. Please reload.
• I also reuploaded today's note; If you don't see a typeset "YES. Trying to schedule one May 16th" under the p1 headline, please reload.  Changes: 
  • Blacker and hopefully readable.
  • Correction p2: points where the gradient of (g-b)^3 is zero.
  • pp8-9: a Lagrange multiplier symbol changed into ζ to avoid "reserved letters in finance"

Last ordinary lecture finished, hooray. Workshop in five minutes. Expect a couple of changes to find their way into the schedule:

• I'll try to arrange for a review session on May 16th. As seminar over the most recent exam. And then whatever you request.
• Next week's workshop - and my consultation hours from then on: I'll try to move them to Tuesday 1215-1400 (where there are no more lectures).
• Problems for the May 3 seminar to be announced shortly.

Question mark because the first bullet item:

• If it suits "everyone", we could schedule an extraordinary exam-problem solving session mid May (exam is the 20th).
  • That decision will affect the final problem set. (Which will include last year's exam if there is no such review.)
• Notes for tomorrow uploaded - in questionable quality, because the usual scanner is down, I will reupload a sharper one tomorrow.
  • Given the short time, I suggest that for the optimization parts you read pages 1-2 and then jump to 5 - because that clarifies exam expectations; that gives you an idea about how much of the rest you want to cover: 3-4 are more on optimization vs 6-7 which are essentially there because it is a nice application to review the implicit function theorem. (8-9 are an example you don't need to stress before the lecture.)
  • The Bernoulli differential equation is covere...

• Seminar problems uploaded
• Concave/quasiconcave functions factsheet corrected (the C1 criteria where I managed to turn everything on its head) and updated with few lines bottom first page, and a fourth page on nonlinear programming.
• The totally optional note on complex numbers had confused "eigenvectors" with "eigenvalues". Corrected.

After Easter: 

• How to destroy Lagrange/Kuhn-Tucker
  • "Application" of this and of concave programs: why you so often see "minimize risk for given expected return" rather than "maximize expectation for given risk" in textbooks.
• The Bernoulli first-order differential equation.

• 2-01, 2-06, 2-07 and 3-05 - for the latter, part (c) is top priority! 
• And: check for quasiconcavity/quasiconvexity the function x + square root{x^2-y}.
• More (and worse!) will be added shortly.

Tuesday after Easter is last ordinary lecture.

First: Seminar problem assignment updated.

I have not yet made a new 2019 note on convexity/concavity/quasi-*. The C^2 characterization is covered already, but we will get as far as to the C^1 characterization and beyond. You should have a look at the note up to the C^1 characterization.

Then, quasiconvex functions: have a look at this plot. The function is not convex, it flattens out: fill with water up high enough, and a line segment from the bottom to the shore will cut outside. But it is quasiconvex: 

• Fill up to level L (fixed, but arbitrary). 
• Look down at the water surface from a bird's view. 
• For this partic...

Suggested agenda (feel free to suggest and request!): review optimal control theory. Which likely would amount to:

• Conditions
• How to handle "the next question after 'state the conditions'"
• As much about differential equation systems as needed
• UPDATE: I could cover for example one of the following problems from the compendium - a bit subject to what you think you struggle at. (Reducing to systems? 9-02/04/10 then. Considering behaviour of p and then inferring from that? 9-13/14 then.)
  • 9-07, which was assigned as least priority for Friday; then Sondre could spend more time on the higher priority questions. 
  • 9-02; a concave problem leading to a diff.eq. system. I did intend to assign this for a later seminar though; if covered tomorrow, I could assign "solve by way of current-value" to get a slightly different one.
  • 9-04 or 9-10 are not unlike 9-02....

Last year's note on convexity & friends is long, and I don't expect you to browse more than a bit of the beginning. Given the short time, I suggest you first browse through the following subset of the first few pages:

. page 1: read the first half page; notice the examples, but need not attempt to show anything

. page 2: read the claim, but not the proof; read the note the last two lines

. page 3

. page 4 (nearly empty - but carries a lot of substance)

. For page 5: draw your "Math 2" concave function of one variable. "Fill the graph with water" and notice that the water is a convex set. Notice that a concave function is a convex turned on its head. Try to visualize both of these properties for a function of two variables.

Here is the note: https://www.uio.no/studier/emner/sv...

Sondre steps in. 

New problem assignment. In part due to the following:

Olav voiced a concern that some of you are struggling right now, due having other term papers concurrently with a hard part of Mathematics 3.
I am open to assigning additional review lectures/problem solving sessions - probably ASAP, I think?  

• How does Thu 28th look in your calendars?  
• Also possible: The workshop the week before Easter?
  (No seminar after lunch Friday before the Easter break ...)

The change in the upcoming problem set hopefully gives you a little bit of air: a mix of old stuff to review, leftovers from yesterday and a little bit of new problems - leaving the harder dynamic programming problems for seminar #9, April 5th (the last seminar before Easter). Likely there will be one postponed for after Easter as well.

I (Nils) am unavailable at lecture time Tuesday, so therefore Vegard Wiborg (whom some of you had in the Math 2 seminars) will step in. He will cover one single topic, moved from the end. Therefore: 

• Tuesday's lecture: Double and multiple integrals.
  • I have uploaded a sketch, in part based on last year's lecture, with some "explaining what's going on" added.
• Tuesday's workshop: I may or may not be back til then. If not, ask someone on the 12th floor to unlock the door, and discuss without me. 
• Wednesday's lecture will start on convex sets and then a proper definition of convex/concave functions. 
• As a consequence of double integrals being inserted, I will update the problem assignment for the 29th. I was worried there would be a dynamic programming overdose, and now I can give you some integrals instead. 
  The DP problems removed, will be reassigned f...

So I made a short note if you only have time to "get the mindset".

My old handwritten lecture note here.

Edit: and if you didn't do problem A off this set, do it now.

Tomorrow Tuesday, we shall

• Spend some minutes on bits and pieces from optimal control theory.
• Proof by induction.
  For a brief walkthrough, read for example: the Wikipedia page up to (but not including) "Example: forming dollar amounts by coins".
• Difference equations. Note uploaded.

Less than an hour, and an update.

Meanwhile I got a question from one of you concerning the so-called p₀ constant that is mentioned in the 2009 exam. It plagues me a bit, because it is open-book exam, and whoever reads FMEA Theorem 9.4.1 / MA2 sec. 12.4 "Degenererte problemer" will see it and maybe use it. OTOH, mentioning it paves room for misunderstanding the first of the following bullet items:

• Important to note: it is not   p(0) or p(t₀). 
• If you use your lecture notes or my just-refreshed upload rather than the above book references, you can ignore every mention of p₀ in any seminar assignment. And it is not exam relevant, at least not in 2019.
• This is more explained in footnote 3 in the note, which I obviously shouldn't have told you to take lightly. 

Another upd...

Next Tuesday, I will spend like, a quarter of an hour talking a little bit more about this. Then we start on difference equations.

I took the old 2018 note on control theory, and updated it to a draft 2019 version (please YELL at me if something needs to be changed):

• I kept page 1 although we haven't gotten to that yet. You may disregard that for the moment.
• page 2: take footnote 3 very lightly.
• page 3: I stuffed in references to the old handwritten notes, with hyperlink (yeah, do not work on paper). 
  The 3rd 2018 note is less needed. 
• Examples now have letter-enumeration, as the handwritten notes have numbers ...
• page 6: in place of yesterday's not-so-good example. 
• page 10: sensitivity moved here, and today's example - in C...

By now, most of you have been working on Math3 problems for a hour and a half, and have probably had enough.  So my suggestion to prepare for tomorrow, will be the following: 

• From last year's lecture 2 notes, read pages 4--6: the current-value formulation. And the phase diagram. 
  I am going to cover that first, to be sure we get it done.

Then I will say something about sensitivity: page 3 the same note. You might read that example too. 

When that is done ... I'll pick another example; possibly from note #3, possibly from the old curriculum note. And then there is not much left I guess.

I realize I should have updated control theory notes, and in particular for tomorrow, as last year's 2nd note starts with "We did not cover" ... Likely last year's typewritten note is better, so the following plan references that one:

• We have already: the problem and the necessary conditions (page 2 on that note).
  ^{(Footnote 3 assumes knowledge of all the catches of Kuhn--Tucker, ignore for now.)}
• Tomorrow: 
  • Review the cookbook (top of page 3)
  • Sufficient conditions (page 3): we have two sufficiency tools. You need to know both.
  • Sensitivity results: the interpretation of p. First half of page...