311 – HW 8

April 23, 2015

HW 8 is due Tuesday, April 28, at the beginning of lecture.

Work in hyperbolic geometry. Given a triangle \triangle ABC, recall that its Saccheri quadrilateral \Box ABB'A' based at \overleftrightarrow{AB} is defined as follows: Let M be the midpoint of \overline{AC} and N be the midpoint of \overline{CB}. Let A',B' be the feet of the perpendiculars from A and B to MN, respectively.

Continuing with the same notation, suppose now that G is an arbitrary point on \overleftrightarrow{MN}, and let H be a point on the ray \overrightarrow{AG} with GH=AG. Show that \Box ABB'A' is also the Saccheri quadrilateral of \triangle ABH based at \overleftrightarrow{AB}.

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MathReviews

April 22, 2015

I will be taking a leave from BSU this coming academic year, and moving to Ann Arbor, to work as an Associate Editor at MathReviews.

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403/503 – HW7

April 16, 2015

This exercise is due Tuesday, April 21, at the beginning of lecture.

Find the Singular Value Decomposition of

\displaystyle M=\left(\begin{array}{ccccc}1&0&0&0&2\\ 0&0&3&0&0\\ 0&0&0&0&0\\ 0&4&0&0&0\end{array}\right).

(I am not so interested in the specific answer, which can be found online, but rather in the process describing how one arrives to this answer.)

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311 – HWs 6 and 7

April 15, 2015

HW 6 is due Thursday, April 16 and HW 7 is due Tuesday, April 21, both at the beginning of lecture.

HW6

Work in hyperbolic geometry.

1. Let \ell and m be two parallel lines admitting a common perpendicular: There are points P\in\ell and Q\in m with \overleftrightarrow{PQ} perpendicular to both \ell and m. Suppose that A,B are other points in \ell with P*A*B, that is, A is between P and B. Let C be the foot of the perpendicular from A to m, and let D be the foot of the perpendicular from B to m.

Show that PQ<AC<BD. That is, \ell and m drift apart away from their common perpendicular.

(Note that \Box PACQ and \Box PBDQ are Lambert quadrilaterals, and therefore PQ<AC and PQ<BD. The problem is to show that AC<BD.)

As an extra credit problem, show that for any number r>0 we can find B (on either side of P) such that BD>r, that is, \ell and m not just drift apart but they do so unboundedly.

2. Now suppose instead that \ell and m are critical (or limiting) parallel lines, that is, they are parallel, and if Q\in m and P\in\ell is the foot of the perpendicular from Q to \ell, then on one of the two sides determined by the line \overleftrightarrow{PQ}, any line through Q that forms with \overleftrightarrow{PQ} a smaller angle than m does, cuts \ell at some point.

On the same side as just described, suppose that C,D are points on m with Q*C*D, that is, C is between Q and D. Let A be the foot of the perpendicular from C to \ell, and let B be the foot of the perpendicular from D to \ell.

Show that PQ>AC>BD. That is, \ell and m approach each other in the appropriate direction.

As an extra credit problem, show that for any r>0 we can choose D so that BD<r. That is, \ell and m are asymptotically close to one another. Do they drift away unboundedly in the other direction?

HW 7

Show that the critical function \kappa is continuous. Recall that \kappa:(0,\infty)\to(0,\pi/2) measures the critical angle, that is, \kappa(x)=\theta iff there are critical parallel lines \ell and m and a point Q\in m such that if P is the foot of the perpendicular from Q to \ell, and PQ=x, then m and \overleftrightarrow{PQ} make an angle of measure \theta in the appropriate direction.

(In lecture we verified that \kappa is strictly decreasing. This means that the only possible discontinuities of \kappa are jump discontinuities. We also verified that \kappa(x) approaches 0 as x\to\infty, and approaches \pi/2 as x\to0. It follows that to show that \kappa has no jump discontinuities, it suffices to verify that it takes all values between 0 and \pi/2, that is, one needs to prove that for any \theta\in(0,\pi/2) there is an x>0 such that \kappa(x)=\theta.)

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403/503 – Another extra credit problem

April 9, 2015

This optional homework is due Thursday, April 16, at the beginning of lecture. We want to verify the details of Jacobi method for diagonalizing symmetric matrices.

Recall that the Givens rotation R(i,j,\theta) has the effect of rotating a vector by \theta radians in the i,j-plane.

Suppose A=(a_{k,l})_{k,l=1}^n is an n\times n real symmetric matrix, and that i\ne j. Let B=(b_{k,l}) be the matrix G^T A G where G=R(i,j,\theta).

  1. Show that if \theta is chosen so that \displaystyle\cos\theta=\left(\frac12+\frac{\beta}{2\sqrt{1+\beta^2}}\right)^{1/2} and \displaystyle\sin\theta=\left(\frac12-\frac{\beta}{2\sqrt{1+\beta^2}}\right)^{1/2}, where \displaystyle \beta=\frac{a_{ii}-a_{jj}}{2a_{ij}}, then b_{i,j}=b_{j,i}=0.
  2. Show that \displaystyle\sum_{k=1}^nb_{k,k}^2=2a_{i,j}^2+\sum_{k=1}^na_{k,k}^2.
  3. Show that \displaystyle\sum_{k,l}b_{k,l}^2=\sum_{k,l}a_{k,l}^2.

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403/503 – Extra credit problem

April 8, 2015

This problem is due April 30 at the beginning of lecture.

Write a program that receives as input a real symmetric matrix A and some tolerance bound \epsilon, and performs the basic QR method to A generating (and printing) a sequence of matrices A_0=A,A_1,A_2,\dots until a stage n is reached where the entries below the diagonal  of A_n are all in absolute value below \epsilon. Once this happens, the program returns the diagonal entries of A_n as approximations to the eigenvalues of A. (Check on a couple of examples that these are indeed decent approximations, at least for A of small size and reasonably small values of \epsilon.)

Most Computer Algebra Systems already have implemented algorithms to find the QR decomposition of a matrix. Instead of using these pre-programmed algorithms, write your own.

(Turn in the code, plus the couple of examples. Comment your code, so it can be easily understood what you are doing along the way. I’m reasonably familiar with Maple, Mathlab, Sage, and most flavors of C. If you are going to use a different language, please let me know as soon as you can, to see whether it is something I’ll be able to verify or if a different language will be needed instead. Ideally, the user can choose the dimension of the input matrix.)

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Woodin meeting

April 1, 2015

IMG_3359

IMAG1270

(Don’t know who to credit for the group picture, but as pointed out by Paul in the comments, it was with David Schrittesser‘s camera. Toast picture by Paul Larson. Toast by Ted Slaman.)

I’m sad I had to miss the meeting, although it was for obvious reasons.

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