J. J. P. VeermanProfessor of MathematicsAffiliate Professor of PhysicsFariborz Maseeh Department of Mathematics
and Statistics

Newspaper (El Pais, Spain) articles August
28, 1988 and March 19, 1989.
Papers starting 1985
Papers starting 1995
Papers starting 2005
Papers starting 2015
Research seminar series at Portland State University (2005, 2006, 2010, 2011)
Research seminar series at Rockefeller University (1996, 1997, 2008, 2009)
A language page.
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Join us in Italy, Summer 2019:
1) July 311,
Pescara, Italy:
I am coorganizing the 6th
Ph.D. Summer
SchoolConference on ``Mathematical Modeling of
Complex Systems".
Please, see here
for more information on the school
and how to apply.
2) July 12,
Pescara, Italy: In collaboration with the summer school and in
preparation for it, some colleagues
from Pescara and I will give a tutorial
on information flow in directed graphs
with some applications.
See here
for information.
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Teaching Winter 2019:
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IMPORTANT
1: All answers on home works and exams must be
justified, even if that is not evident from
the phrasing of the question. Answers without
justification will receive partial credit at best.
IMPORTANT 2: Before
turning in exams or HW's, write your first plus last name in the
top right corner
of each sheet you turn in (even if you
staple them together)!
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MTH 435/535, Topology
Here you can find the syllabus.
All assignments, home works, and exams are announced in
class.
ASSIGNMENTS: Tues, Jan 08: Kaplansky 4.1
and 4.2, unstarred exercises.
Tues, Jan 15: Kapl 4.3 and 4.4, unstarred exercises.
Tues, Jan 22: Kapl 5.1 unstarred.
Tues, Jan 29: Kapl 5.2 unstarred.

Armstrong:
Tues, Feb 12: 1.6: 17, 10, 1417, 23.
Tues, Feb 19: 2.1: 2, 3, 4, 9, 11, 12; 2.2: 13, 14, 16, 17, 19;
2.3: 2226 (you'll find nice answers on the web).
Tues, Feb 26: Ch 2: 2733. Ch 3: 1,2. Show that intersection of
nested closed sets is nonempty (Kapl. pg 87).
Ch 3.3: 512.
Tues, Mar 05: Ch 3: 20  23, 25, 30  34, 37  44; Ch 4: 1, 2,
3, 5, 7.
Tues, Mar 12: Ch 4: 13, 16, 17, 18, 19.
Tues, Mar 19: Ch 4: 2629, 30a.
EXAMS:
Midterm1:
Thurs, Feb 07. Kapl. Chapters 4 and 5.
Midterm2: Tues,
Feb 26. Turn in takehome exam: typewritten proof of Euler's Thm.
(Please note: Must also include the additional proofs done
inclass!)
Final: Tues,
Mar 19: 17:3019.20. All material
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Here you can find the syllabus.
All assignments, home works, and exams are announced in
class.
ASSIGNMENTS: Tues, Jan 08: 
Tues, Jan 15: 1a) Show exp(tA).exp(sA)=exp((t+s)A). b) Give
example to exp(A).exp(B) not equal exp(A+B).
2) x' = ax+byx(x^2+y^2)
y' = bx+ayy(x^2+y^2)
a) Transform to polar coords, b) Identify the bifurcation in
the r coordinate, c) Determine angular velocity.
3) x''+bx'+cx=d cos(omega t)
Give a complete analysis of this system. In particular, discuss
bifurcations, if any.
4) Extra Credit: get the correct form of Q(k,t) on page 203204.
5) Let T be the unit square with sides identified to give a torus.
Let c(t) be the constant (nonzero) velocity orbit starting at O.
Discuss omega(c) as function of c'(0).
6) Extra Credit: On the torus (see probl 5), it is possible that
the omegalimit set is fractal.
Look up Cherry flows.
Tues, Jan
22: 1) Extra Credit: Explain how Van Der Pol modeled the heartbeat
with equation (10.3).
2) Explore the behavior of eqn (10.56) when mu crosses from
negative to positive.
3) In eqn (11.2), compute "x bar" and "y bar" and determine effect
of nonspecific insecticide.
4)
Linearize eqn (11.2) at the "interior" fixed point (figure 11.4).
5) In eqn (11.3) case 4, find a Lyapunov fn like V to show that
the (local) attractor of case 4 attracts all initial
conds in the interior of the pos. quadr.
Tues, Jan 29: 1) Extra Credit: Find more general criterion on the
interaction matrix st Thm 11.4.2 holds.
2) Give an explicit 3dimensional example of eqn (11.6).
3) Apply
Hofbauer's thm to the above eqn and find the associated replicator
eqn.
4)
Investigate (numerically or otherwise) if your system (of probl 2)
has an evolutionary stable state.
5) Assume all letters of the alphabet occur equally often (frequ.
= 1/26). Find an instantaneous
binary encoding the alphabet and show that it satisfies
both Shannon theorems.
6) In the case of problem 5), what are the codes that satisfy the
first formula on pg 144?
7) Same
question as 5), but now assume that A occurs with frequ = 1/2, and
all other 25 letters with
frequ = 1/50.
Tues, Feb 05: 1) Prove that d(x,y) on page 247 is a metric.
2) Prove
that sigma on page 248 is continuous.
3) Show
that h on page 249 is continuous.
4) Let
F=(01101) and show the 4step shift (binary strings). Compute the
topological entropy.
5) Figure out the connection between topological entropy and the
entropy in Chapter11.

HofbauerSigmund.
Tues, Feb 12: 3.3: 1, 4; 3.4: 2, 3, 4; all of 4.3 and 4.4 and 4.5.
Tues, Feb 19: Ch 5: 5.5.25.
Tues, Feb 26: Ch 6: 6.1.1, 6.3.1, 6.4.2, 6.4.3.
Tues, Mar 05: Ch 7: 7.4.13, 5, 6.
EXAMS:
To be discussed in class.
General Announcement
for Students:
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In most of my classes you will be either strongly encouraged,
or even obliged to turn in your HW in *.pdf format based on LATEX.
Here is a website where LATEX is explained:
http://www.ctan.org/texarchive/info/Math_into_LaTeX4/