Seminar problems 2019

Seminar #1, January 30:

If the compendium and old exam problems below are too hard at this stage, then these should be simpler:

• FMEA problem 1.2.4, LA problem 7.1.4.
• FMEA problem 1.3.1(a),(b),(c)/LA 7.4.5(a),7.4.1(a),(b)
• FMEA problem 1.4.1(a),(b)/LA 8.1.1(a),(b)

Over to the real thing. Some compendium problems: 

• Rank etc.: 1-12
• Eigenvalues/-vectors & friends: problems (f), (h), (i), (j) of this compulsory term paper problem set from MatNat: http://www.uio.no/studier/emner/matnat/math/MAT1001/h16/oblig1_eng.pdf
• Both of the above: 1-10.  For 1-10 (b), you can use tools like Wolfram Alpha (example) if you do not remember polynomial division.  But be sure to learn it ... 
• Time thief: 1-14 (lot of work, maybe lower priority). 

And part of some (not too easy) exam:

• 2008 problem 1 parts (a),(b).

Seminar #2, February 8:

Updated after the Tuesday 29th lecture: hopefully final. please e-mail the seminar leader your priorities.

• True or false?
  "In order to find the general solution of the differential equation x' + a(t) x = b(t), it suffices to find (I) one particular solution y(t), and then a nonzero particular solution u of x' + a(t) x = 0; then the solution we are looking for will be x(t) = y(t) + Cu(t), C any constant." 
• 1-03, 1-05, 1-06 and 1-07 ("mutually orthogonal" means that the dot product of any two distinct vectors in the set, is zero).
• Is it so that an eigenvalue for A must be an eigenvalue for the transpose A'? (No symmetry assumption made.)
• Let A be symmetric. Show that the value of the problems min/max x'Ax subject to x'x = 1, is, respectively, the smallest and largest eigenvalue.
• Let A be negative definite, let p be an integer and M be the power A^p of A. 
  • Show that if p is odd and positive, then M is negative definite def;
  • Decide if the same thing holds if p is even and positive
  • What about negative odd powers? (That is, odd powers of the inverse of A - by the way, is it obvious that the inverse exists?)
• Exam 2009 problem 1.

If you need easier book problems to start with (not to be coverered except possibly by popular demand!), just start with the problems from the beginning of the eigenvectors section (FMEA 1.5 (blue ed.), LA 10.1) and from the beginning of the quadratic forms section (FMEA 1.7 (blue), LA 10.5) - omit the linear constraints sections. I see that those problems have quite different enumeration, so if you want them covered: take a cell phone photo (and maybe reduce size - a good phone camera takes pics that take time to up/download over a mobile network) and e-mail, so that it is clear which one it is.. 

Seminar #3, February 13:

It might be too much - depends on how well you pick up speed at solving differential equations (in the end, you will probably be able to solve them quickly). In case "too much": postpone "from the bottom". Up to the three last ones (starting with 6-12) are candidates to be postponed. 

• Use integration by parts to find an antiderivative of t cos t.
• 1-13
• 6-01
• 6-02. In (b), try the hint and fit coefficients!
• 6-08: "do enough to catch the point" (how far can you get without Mathematics 3?)
• 6-10. It is not curriculum to solve this equation from scratch, but given the hint it is curriculum to be able to construct the general solution of the homogeneous. For (b) ... make an educated guess!
  This problem is in part given to give an upper bound on "how bad could it possibly get with non-constant coefficients?"
• 6-12. Requires the Leibniz rule. Equilibrium value means a stationary state means a constant solution. (c) requires Thursday's lecture. 
• 7-01. Requires Thursday's lecture.
• Exam 2008: rest of problem 1; part (d) will require Thursday's lecture.

Seminar #4, March 01:

• everything you didn't have time for for Seminar #3. 
• Compendium 6-15, 7-01, 7-02, 7-04
• Problem 4 from https://www.dr-eriksen.no/teaching/GRA6035/2011/final-2011-12.pdf
  This contains "hard" elements!
• The problems at https://www.uio.no/studier/emner/sv/oekonomi/ECON4140/v19/additional-problems-for-the-march-1-seminar.html
  Don't expect all of these to have top priority.

Seminar #5, March 8:

• Leftovers from the previous seminar. I was told it was a bit too much.
• Find and classify the equilibrium point of the system
  x' = 1 - exp(x-y),  y' = -y.
  (This example is from the book under Olech's theorem, a global test which is not curriculum.  You are only asked about local (asymptotic) stability.)
• 7-05
• 8-01
• 8-06
• 9-05. Part (d) requires optimal control lectures.

Seminar #6, March 15:

• 9-01 
• 9-03
• 9-12
• The full exam 2009. 
  I earlier suggested to allocate a 3hr slot for a "mock exam" in the Tuesday the 12th workshop, but at this stage you will probably not have gotten up to full speed. So, maybe not. 
  Which means, the workshop would not be a silent area ...

Seminar #7, March 22:

Updated March 15.

• Optimal control: 9-17. Also: If x(0) were changed from 0 to z (near 0), approximately how much would the optimal value change?
• Induction problem 1. Use induction to prove these classics for all natural n: 
  • (a) \(1+\ldots+n=(n+1)n/2\)
  • (b) \(1+m+\ldots+m^n=\frac{1-m^{n+1}}{1-m}\)when a is not 1. You shall not multiply both sides by (1-m), but there will be a term that you want to multiply by (1-m)/(1-m)
  • (c) \(\mathbf I+\mathbf M+\ldots+\mathbf M^n=(\mathbf I-\mathbf M)^{-1}(\mathbf I-\mathbf M^{n+1})\)when 1 is not an eigenvalue of the square matrix M. Again, do not multiply both sides by I-M.
• Induction 2: Write parts (a) and (b) of Induction problem 1 as linear first-order difference equations \(x_{n+1}=a_n x_n+ b_n\) with x₀ = 1.
  (That is, for each of those two problems: find a_n and b_n.)
• Induction 3: Combine with difference equations. Suppose that \(\mathbf x_0\) and each \(\mathbf b_t\) are linear combinations \(\mathbf x_0=\mu\mathbf u+\kappa \mathbf v\) resp. \(\mathbf b_t=\alpha_t\mathbf u+\beta_t\mathbf v\) of two eigenvectors of \(\mathbf A\) (note: the same two eigenvectors everywhere), and define \(\mathbf x_1,\ \mathbf x_2,\ \dots\)recursively by the difference equation \(\mathbf x_{t+1}=\mathbf A\mathbf x_t+\mathbf b_t\).
  Use induction to show that each \(\mathbf x_t\) is a linear combination \(\zeta_t\mathbf u+\sigma_t\mathbf v\).
• Difference equations: compendium 10-01
• Difference equations: compendium 10-05
• Differential eqs vs differential equations 1:
  • Find the general solution of (A) \(16 \ddot x+24\dot x + 5x = 90\)
  • Find the general solution of (B) \(16 x_{t+2}+24x_{t+1} + 5x_t = 90\)
  • Is (A) stable/asymptotically stable? Is (B) stable/asymptotically stable? (If we have not covered stability of difference equations: the definition for (asymptotic) stability of a linear difference equation is the same as for a linear differential equation system: all particular solutions of the corresponding homogeneous equations stay bounded as time grows (stability), and for asymptotic stability: converge to zero as t grows
• Differential eqs vs differential equations 2: Same questions as previous except with
  (A) \( \ddot x-\tfrac{1}2 \dot x+ \tfrac5{16} x = e^{-t}\) and (B) \(x_{t+2}-\tfrac{1}2 x_{t+1}+\tfrac5{16} x_t = e^{-t}\)
• Induction problem Whiskey.Tango.Foxtrot: In a textbook by Hungarian mathematician George Pólya, he gave this example of a flawed induction "proof" that all horses have the same colour. Find the flaw:
  • Once the induction step is completed, we only need to point out that in any group of size N=1 - i.e., a single horse - there is only one colour.
  • There are n horses in total. We use induction on n:
    1. Suppose true for any group of N horses (i.e. any N horses have the same colour). Consider a group of N+1 horses.
    2. This group is formed by taking a group of N horses - by hypothesis, all of the same colour - and augmenting by horse number H = N+1.
    3. Replace one horse in that group of N by horse number H. That group still has N horses, which by hypothesis have the same colour.
    4. Thus, horse H must have the same colour as the others in the group of N, hence the same colour as all the others (because the replaced one had the same colour too).

Optional "extra problems" (as if Polyá's isn't optional enough):

• Fix numbers a, b and z. Let x_-1 = 0 and x₀ = z. Define recursively x_t+1 = a(x_t + x_t-1) + b for t = 0, 1, ... (Yes, this is second order, though it is deliberately written a slight bit different than you might be used to ...)
  Then each x_t depends on a, b, z and t: \(x_t=w(a,b,z,t)\). Use induction to show that 
  • w has the form  \(w(a,b,z,t)=p_t(a)z+(1+q_t(a))b\), where
  • \(p_t\) and \(q_t\) are polynomials of degree t or less, with no constant term (i.e., \(p_t(0)=q_t(0)=0\)).
• Now do the same in higher dimension. That is: Given two vectors \(\mathbf z\) and \(\mathbf b\) in \(\mathbb R^n\) and an nxn matrix \(\mathbf A\). Define \(\mathbf x_t\in \mathbb R^n\) recursively by \(\mathbf x_{-1}=\mathbf 0\) and  \(\mathbf x_0=\mathbf z\) and \(\mathbf x_{t+1}=\mathbf A(\mathbf x_t+\mathbf x_{t-1})+\mathbf b\). Use induction to show that for every natural number t,  \(\mathbf x_t\) is of the form \(\mathbf x_t=p_t(\mathbf A)\:\mathbf z+(\mathbf I+q_t(\mathbf A))\:\mathbf b\), where \(p_t\) and \(q_t\) are polynomials of degree t or less, with no constant term (i.e., \(p_t(\mathbf 0)=q_t(\mathbf 0)=\mathbf 0\)).

Seminar #8, March 29:

March 23rd: From your feedback by way of the contact student, I understand that you are worried you will be able to catch up at this stage. This week's problems are a mix of new and old, with the heaviest focus on dynamic programs moved to next week.
^{(I am not sure if I should even write "Your feedback matters!", because these days that seems to mean "Give us all your privacy!".)}

• Reassigned from last time: Differential eqs vs differential equations 1:
  • Find the general solution of (A) \(16 \ddot x+24\dot x + 5x = 90\)
  • Find the general solution of (B) \(16 x_{t+2}+24x_{t+1} + 5x_t = 90\)
  • Is (A) stable/asymptotically stable? Is (B) stable/asymptotically stable? (If we have not covered stability of difference equations: the definition for (asymptotic) stability of a linear difference equation is the same as for a linear differential equation system: all particular solutions of the corresponding homogeneous equations stay bounded as time grows (stability), and for asymptotic stability: converge to zero as t grows
• Reassigned from last time: Differential eqs vs differential equations 2: 
  Same questions as previous except with
  (A) \( \ddot x-\tfrac{1}2 \dot x+ \tfrac5{16} x = e^{-t}\) and (B) \(x_{t+2}-\tfrac{1}2 x_{t+1}+\tfrac5{16} x_t = e^{-t}\)
• Linear algebra:
  True, false, or "no way to know in Mathematics 3":
  "A 2019x2019 matrix always has a real eigenvalue."
• Linear algebra and induction: 
  Compendium 1-08, and furthermore: guess the tth power At, and prove it by induction  
• Linear algebra, induction and a matrix difference equation:
  Compendium 1-04, where you in (b) should do as follows: 
  • Suppose first that x₀ can be written as a₀u + b₀v, where u and v are (linearly independent) eigenvalues of A, and check that the approach of the "Induction 3" problem from last time works. Use this to find formulae for a_t and b_t. 
  • Then show that x₀ can be written as a₀u + b₀v, and use that to solve.
• Dynamic programming: Give at least the first of these high priority, so you have done at least one before the harder problems for next week
  11-04 and 11-07 
• Integrals: Again, prioritize the first.
  4-05 and 4-08
• Least priority, could be reassigned: optimal control theory review,
  9-07

So least priority to cover will be 

• Everything easy if anything (the true/false question could maybe be done in a minute's handwaving ... maybe)
• 11-07, 4-08 and 9-07.

Seminar #9, April 5th

Updated March 31st (the problems starting after the 5150 difference equation thrown in, the stochastic dynamic program relegated)

• 10-03
• 11-01 (here you have to do the form yourself, unlike 11-04 and 11-07 for last time)
• Exam 2008 problem 2. Warning: 2nd order conditions!
• I give this as a "read the text and extract the problem on Math3 form and then solve it"; if you want a "don't bore me with text, give me the equation to solve" spoiler, look here.
  Look at the difference equation given in problem 3 (a) in this exam in ECON5150. Here, the unknown function is v, while  p∈(0,1) and K are constants. The running index is called "i" (it isn't time in this application).
  • Is there any p∈(0,1) such that the equation is globally asymptotically stable? (Do not worry so much over the stable-but-not-asymptotically case.)
  • Find the general solution. Note, constant solution will be a particular solution to the homogeneous, so for a particular solution to the inhomogeneous, try L_i except when p=1/2; then try a quadratic Q_i²
  • Find the particular solution which satisfies v₀=v_n=0. 
    (The Department has discontinued Mathematics 4. If you think this application looks interesting, then try STK2130 - be aware though, that for the time being 2*** courses may not be useful towards your degree even though the content is at a level we used to assign to PhD candidates.)
• (this and the following added March 31st).  Consider the dynamic programming problem \(\displaystyle J_{t_0}=\max_{u_t\in[0,1]}\Big\{2^{t_0-T}Q\cdot \ln x_T+\sum_{t=t_0}^{T-1}2^{t_0-t}\big(\ln x_t+\ln u_t\big)\Big\}\) where \(x_{t+1}=(R-u_t)x_t\) and where Q and R>1 are positive constants.
  • Show by induction that the value function is of the form A ln x + B, where A and B depend only on (T-t).
  • Let now T be positive infinity (it does not matter whether we consider limit or "proper infinity").
    • State the Bellman equation
    • Guess a form with two constants and find equations for them.
• 4-09
• Requires Tuesday's lecture: Let \(f_1,f_2,\dots,\) be convex functions defined on a common domain S. Use induction to show that all the functions \(F_n\)defined by \(F_n(\mathbf x)=\max\{f_1(\mathbf x),f_2(\mathbf x),\dots,f_n(\mathbf x)\}\) are convex.
• (Tricky!) In the previous problem, induction was mandated - but otherwise one could do it without. In this problem, you cannot use induction.
  Fix a convex set S and an arbitrary index set J. Assume that \(f_j\) is defined on S and convex, for every index \(j\). Define F by F(x) = the smallest real number that is \(\geq f_j(\mathbf x) \) for all \(j\in J.\) (That is, go as large as you have to to dominate all the fj, but no further. We call this least upper bound "sup". For example, if S is the interval (0,1] and J is the interval (2,3) and each \(f_j\) is the function \(f_j(t)=jt\), then F(t)=3t; although none of the \(f_j\) is as large as 3t, then you have to go that high to get something that is \(\geq \) all the \(f_j(t) \) output values.) Problem for you: 
  • Is it so that this F defined by \(F(\mathbf x)=\sup_{j\in J}f_j(\mathbf x)\) is a convex function? If not, what additional assumption has to be made?
    (Trick question, I admit; if you think you have shown that F is a convex function, then you have done what I intended to test; if you can address the "nitpickers' issue" as well, then congratulations).

Seminar #10, April 26th

This set is a mix of "easy", "easy if you find the right way to do it and could take a lot of time if not", and ... admittedly quirky. 

• [CORRECTED - TWICE - BEFORE 1632 @ THE WORKSHOP, MISSED THE DISCOUNT FACTORS \(\beta^{t-t_0}\)AND \(\beta^{T-t_0}\). ^{sorry ... and "arrrgh" ...}]
  Let \(\beta\in(0,1)\) and consider the dynamic programming problem \(J_{t_0}(x)=\displaystyle\max_{u_t\geq 0}\Big\{\beta^{T-t_0}(x_T+\sqrt{x_T})+\sum_{t=t_0}^{T-1}\beta^{t-t_0}\big(u_t+\sqrt{u_t}\big)\Big\}\) 
  where \(x_{t+1}=\frac1\beta(x_t-u_t)\) , starting at \(x_{t_0}=x>0.\)
  • Calculate J_T-1 and J_T-2. 
  • By now you should be able to guess a form of the value function J_T-t(x) up to some A_T-t that does not depend on x; prove this by induction. If you cannot guess, see cheatnote a couple of items further down.
    • If you chose a nice form as per the cheatnote, the difference equation for A should be "easy" to solve as well!
  • Let T go to infinity. State the Bellman equation for this problem.
  • Guess and verify a form for the value function. (Cheatnote: this form is probably most convenient!)
• 2-03 without calculating determinants (might not be prioritized)
• 2-01
• 2-06
• 2-07
• Consider the function \(f(x,y)=x+\sqrt{x^2-y}\) (defined whereever the square root is.)
  • What do the level curves look like? 
  • Is the function quasiconcave/quasiconvex on its domain? If not: Where is it quasiconcave/quasiconvex?
• 3-05 with part (c) being top priority. 
• Look at 3-12 and 3-13 without solving them: do we for any of these problems have tractable tools to tell up-front 
  • Whether there must be a point satisfying the Kuhn-Tucker conditions?
  • Whether such a point, if exists, must in fact solve the problem?
• [Requires the lecture Tuesday after Easter. Low priority.] Consider the problem \(\min\ (x+1)^2+(y+1)^2\quad \text{s.t.}\quad y\geq \sqrt{x-1}\) 
  • Find all the points that satisfy the Kuhn--Tucker conditions, if any.
  • Solve the problem graphically.
  • Surprised? Explain.

Seminar #11 

Updated April 28th.

I am a bit in doubt on whether to give you two full exams (more exam directed, obviously!) or only one (saving one for later). I give two alternative sets, and urge you to mail your priorities to Sondre - who suggested "Alternative 1". The downside to that is ... it could be quite a lot - then on the other hand, there are no more lectures (in this course).
Leftovers will be covered upon request on the 16th, and a solution will in any case be produced.

• In any case: The full 2015 exam.
  • Problem 2 is "the hardest of its kind given this far", and especially since you have not been assigned anything "simpler of this kind", do not despair it that set takes longer than three hours.

Alternative 1: one more set. Could be a lot ...  

• The full 2015 exam, and the full 2012 exam.

Alternative 2: bits and pieces and drill and ...

• The full 2015 exam
• Leftovers from last time! In particular, if you didn't do the last "low priority" problem from last time: do it!
• Compendium 5-14. (Not straightforward.)
• The so-called trace norm of a matrix is the function \(\ell(\mathbf A)=\sqrt{\textsf{tr} (\mathbf A\mathbf A^{\textsf T})}.\) That is, you multiply it by its transpose, sum up the main diagonal, and take the square root. Questions:
  • Verify that if the matrix is a row vector or a column vector, the trace norm equals the "ordinary" norm.
  • Describe in words what \((\ell(\mathbf A))^2={\textsf{tr} (\mathbf A\mathbf A^{\textsf T})}\) is.

And, extra drill problems if you feel that you need something on quasiconcavity/quasiconvexity or dynamic programming:

• Consider the function  \(f(x,y)=x+\sqrt{x^2-y}\) from last time. The function g(f(x,y)) has Hessian determinant equal to \(-\Big(\frac{g'(f(x,y))}{2(x^2-y)}\Big)^2\). Why does this prove that f cannot - on any nonempty open set - be written as a transformation of any concave function nor of any convex function?
• Application: the wording of this problem makes it unsuitable for an exam, but if you know your theoretical micro, then you should be able to translate back and forth.
  An economy has agents indexed by k. An allocation can either be given as a matrix X where column number k is agent k's holdings of goods indexed by row number. Then the total amount of goods in the economy is the vector X1, where is the vector 1 of ones. A reallocation V of X is just a matrix so that V1 = X1. 
  • If you are comfortable with using matrices this way:
    Fix an allocation X. Show that the reallocations form a convex set.
  • If you are not comfortable with convex sets of matrices - only of vectors - then you can stack the columns atop each other to a vector of order mn, so that the first n elements x₁,...x_n are the first column of X, the next n ones are the next etc. The total amount of each good in the economy would then be Ax where A is the n x mn matrix with row i as follows: zeroes except at i, i+n, i+2n, ..., i+(m-1)n. 
    A reallocation would then be a vector v such that Av = Ax. Show that these form a convex set.
  • In both cases: suppose that all agents have convex preferences, that is, each agent k has a quasiconcave utility function f_k . Fix an allocation X or x. 
    Show that the set of Pareto-improving allocations, is convex. (That is, the set of v such that no agent k experiences f_k(v) < f_k(x) .)
    Show that the set of Pareto-improving reallocations, is convex.
• This problem is beyond Math 3, so hints - hopefully sufficient - are given; do not stress it more than what you think is useful to learn Math 3-level dynamic programming. 
  In the above 5150 exam, problem 1 is stochastic dynamic programming. In the DP equation, replace J_t+1(...) by the expected value \(\mathsf E[J_{t+1}((x_t-u_t)V_t)]=\frac12\cdot\big[J_{t+1}(0) + J_{t+1}(4(x_t-u_t))\big]\)
  ^{(Note: In theory, it is generally conditional expectation, conditioned upon what you know at time t. But in that problem, the shocks are iid.)}
  • Here is your task: Show that the dynamic programming equation for this problem, reduces to the one for last week's DP problem.
  • Note: In Math3, I started at running time t₀ rather than at a fixed time. You are free to do that as well. But, remember that the 1/2 in the expected value is not the same as the 1/2 in the discount factor, so you need one more ...

For your exam preps: full exam sets 

This year I assigned fewer full exam sets for seminars, in order not to "spend" what you would use for exam preps. You have many sets that could be left unseen and then you could simulate an exam for yourselves. That is also one reason why 2015 was assigned for the last seminar; given the focus this semester, it could take you too much time to be a realistic benchmark.

In general, I suggest you start with the more "recent" exams (... since 2006, there has become more weight on Kuhn--Tucker in Mathematics 2 and less in Mathematics 3).
Some remarks:

• Some of these sets have that pesky p₀ constant in optimal control theory. As said before, not related to p(0). If you don't want to be bothered, just delete every mention of  p₀. You will not be asked to handle it on the exam.
• Weighting. From 2011 on, I tried to keep the Math 2 rule-of-thumb that unless otherwise stated, each letter could carry equal weight (deviations would largely benefit the majority).
  • 2016 and 2017 had other weights indicated though.
  • 2014 had an "experiment" that I have seen done elsewhere. In problem 4:
    (c) «Bonus» question: this part will be deleted (zero-weighted) if that benefits your grade.
    That meant that 4(c) blank and everything else perfect, would give 100 percent score - but those who did 4(c) could have the average calculated with or without it and the best average applying. Reality was, it was largely skipped.
  • 2004 had a similar "Difficult. Can be skipped." (I do not know how that was graded.)
• All sets are two pages except for one set, that has a third page with a hint.
  (That could happen again, but at this stage in your study you should make sure to check all the pages and backsides for ink ...)
• 2018 will be covered on May 16th.

For reference:
In 2016 and 2017, the following compendium and old exam problems were assigned (in addition to a few problems typed).
Problems assigned only 2016: normal font. Only 2017: oblique font. Both: boldface.
Strikeout: assigned this far in 2019.

Compendium problems:

• (1-01,) 1-03, 1-04, 1-05, 1-06, 1-07, 1-08, 1-11, 1-12(a), 1-12(b,c), 1-13, 1-14
• 2-01, 2-02, 2-03, 2-06, 2-07
• 3-03, 3-05, 3-07, 3-12, 3-13, 3-14, 3-21
• 4-01, 4-02, 4-04, 4-05, 4-08, 4-09, 4-10
  (note however, the Leibniz rule is now Math 2, obsoleting a subset of these.)
• 5-06, 5-14
• 6-02, 6-08, 6-10, 6-11, 6-12, 6-15 (and generally, start from the beginning of section 6 to drill 2nd order diff.eq's)
• 7-01, 7-02, 7-04, 7-05
• 8-01, 8-06
• 9-01, 9-03, 9-02, 9-05, 9-07, 9-12, 9-17
• 10-03, 10-04, 10-05
• 11-01, 11-04, 11-07

Exam problems

• Exam 2008 #1 (ab); (c), (d). (optionally: problem 2).
• Exam 2009, full set
• Exam 2011 problems 1d and 3. And 4.
• Exam 2012, full set
• Exam 2014
• Exam 2015 all problems