J. J. P. VeermanProfessor of MathematicsAffiliate Professor of PhysicsPortland State University1825 SW Broadway, Portland, OR 97201, USAFMH, rm 464Bemail: veerman@pdx.edu
Website:
http://web.pdx.edu/~veerman/ |
Newspaper (El Pais, Spain) articles August
28, 1988 and March 19, 1989.
Papers starting 1985
Papers starting 1995
Papers starting 2005
Papers starting 2015
Papers starting 2020
Research seminar series at Portland State University (2005, 2006, 2010, 2011)
Research seminar series at Rockefeller University (1996, 1997, 2008, 2009)
A language page.
Short Bio and
Description of Research
Pescara 2019 summer
school and conference on
Mathematical Modeling of Complex Systems
In the summer of 2019, I
participated in a summer school and conference
in
Mathematical Modeling in Pescara, Italy. I gave a mini-course
on the theory
of directed graphs in
the summer school: Part 1,
part2,
part 3,
and part
4.
I also served as guest
editor of the Proceedings of the
conference. A
description
of the meeting and links to
the papers
presented can be found here:
https://www.euro-acad.eu/news?
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Lecture Notes on
Number Theory
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=========================================================
Teaching FALL 2020:
=========================================================
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)!
===============================================================
MTH 449/549,
Number Theory
Here is a flyer, the syllabus,
and three months
of free lecture
notes
for a newly designed course in number theory.
This a sequence of two
courses MTH 449/549 and MTH 410/510
(fall 2020 and winter 2021).
The
material is intended to give a
general overview of all branches of number theory
with an emphasis on
more geometric proofs.
In due course, we hope
to follow this up with a final course
discussing some of the many
applications of number theory.
ASSIGNMENTS:
Tues, Oct
06:
Ch 1.
Tues, Oct 13:
Ch 2.
Tues, Oct 20:
Ch 2.
Tues, Oct 27: Ch 3.
Tues,
EXAMS PLANNED (may be modified in class):
Exam 1 due: Thurs, Nov 05, 14:00 hrs.
===============================================================
MTH 621, Advanced Differential Equations
Here is a flyer and
the syllabus for my advanced
course in dynamical systems MTH 621/2/3. The emphasis this year
will be on surveying the wide range of this branch of
mathematics and its applications, rather than
probing deep theorems. We will use a great -- and very
accessible -- text written by the eminent
mathematician S. Sternberg plus some other resources to be
determined during the course.
This is three term sequence. In Fall and Winter, we will go
through the entire book by Sternberg.
Spring of 2021 will most likely be organized as a 'topics'
course. This will in part depend on students'
interests. Among the possibilities are topics in Ergodic Theory,
Number Theory, Quantum Mechanics,
and so forth.
To all dynamics students I strongly recommend MTH 610, Directed
Networks, as an excellent
companion course to this sequence. It will be taught in Spring
2021. A description can found below.
ASSIGNMENTS:
Tues, Oct 06:
Study Chapter 1. Look at the Julia Set for Newton's method
applied to z^3-1:
https://en.wikipedia.org/wiki/Julia_set#/media/File:Julia-set_N_z3-1.png
And
do one plus 2 of the following:
1)
Perform a Newton's method in 3 dimensions.
2)
Optimize Sternberg's proof of the Newton's method (quadratic
convergence).
3) Show that the secant method has order golden
mean convergence.
4)
Work out the renormalization arguments on pg 27--29.
In
particular, argue that convergence to 1 suggests a large
cluster.
Tues, Oct 13: Study
Chapter 2, pg 33-56. Do at least one in great detail:
1)
Plot the attractors of x->mu.x(1-x) as in Figure 2.9, but
for mu in [3.5,3.8].
and
numerically determine for which mu the orbits of period 2, 4,
8, 16, and 32 are superstable.
2)
Write a different proof for the related pitchfork bifurcation
(what is the relation?).
Tues, Oct 20: Study Ch 2. Steven
Wolfram wrote a nice obituary for Feigenbaum (who passed
in June 2019):
https://writings.stephenwolfram.com/2019/07/mitchell-feigenbaum-1944-2019-4-66920160910299067185320382/
Study these notes. Do at least one of:
1)
Numerically determine Feigenbaum's constant (as outlined in
Sternberg, see
also:
https://pages.physics.wisc.edu/~snc/papers/coppersmith%20am%20j%20phys%201998.pdf).
2)
Set g(y) = 1+ay^2+by^4 and try to solve eqn (2.13) for a
and b.
3)
Give a full proof for the Sarkovski theorem. It is
mandatory to include explicit
references
to all your sources for this home work.
Tues, Oct 27: Study Ch 3 and Ch 4. Do 2 out of the following:
1)
Do this HW.
2)
Draw (numerically) the Mandelbrot set.
3)
Show that conjugation of f : R -> R by a diffeo does
not change the
eigenvalues
at a periodic fixed
point.
4)
Fig 3.6 Sternberg: How long does F^n(x) stay in the
``bottleneck'' as fn of the
distance
of F to the diagonal.
Tues, Oct 30: Study Ch 4. Do
the following HW's:
1)
Prove that closed nested intervals in R have non-empty
intersection.
2)
Show that if the map i on pg 100 is not 1-1, then there is an
interval
J and a K>0 such that:
for
all n>0, Q^n is monotone and | DQ^n (x_n) | < K
for some x_n in
J.
3)
Give an example of a map whose periodic orbits are dense, but
is not
chaotic.
4)
Give an example of a map with sensitive dependence, but which
is not
chaotic.
5)
Give an example of a map which is top. trans., but which is
not chaotic.
6)
Prove Thm: If f and g have attracting fixed points at the
origin, then f and
g are locally conjugate.
7) Prove Thm 4.6.1 for c < -2.
EXAMS
PLANNED (may be modified in class): Exam1: Due Tue Oct 27,
17:15hrs.
Turn in 1 problem for each of Ch 1, 2,
and 3.
===============================================================
Other Teaching 2020-2021:
=========================================================
Spring
2021: DIRECTED
NETWORKS ::
Here
is a flyer containing references to
freely available lecture
notes for this
newly designed
course
describing theory and
applications of directed graphs (MTH
610, spring 2021). This is
a very current area with
applications from data science to flocking,
finance, chemical
networks
and others. In spite of its
current
relevance, good textbooks are
still very rare. So we developed our own course,
and are writing a textbook
based on this course.
===============================================================
General Announcements for
Students:
===============================================================
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/tex-archive/info/Math_into_LaTeX-4/