Lecture notes

Series/conference	Date	Title	Materials
Two Sigma Investments	Jan. 2012	Markov Jabberwocky: Through the Sporking Glass	Abstract, slides.
UA Math Dept. Graduate Student Colloquium	Apr. 2010	Notes on a non-academic job search	Abstract, slides.
Job talk	Mar. 2010	Markov chain Monte Carlo methods and random spatial permutations	Slides.
University of Georgia Center for Simulational Physics 2010 Workshop	Feb. 2010	A worm algorithm for random spatial permutations	Abstract, slides, manuscript.
UA Math Dept. Graduate Student Colloquium	Feb. 2010	Tensorama	Abstract, slides.
AMS/MAA Joint Meetings, San Francisco	Jan. 2010	MCMC methods for random spatial permutations	Abstract, slides.
UA Math Dept. Software Interest Group	Oct. 2009	Python for the working (or aspiring) mathematician	Abstract, slides, hello.py, lorenz.py, plot2.py, projdown.py, tabutil.py, all-in-one.py.
UA Math Dept. Math. Physics Seminar	Sep. 2009	Critical behavior for the model of random spatial permutations	Abstract, slides.
UA Math Dept. Graduate Student Colloquium	Aug. 2009	Markov Jabberwocky: fesh, excenture, and the like	Abstract, slides, lay-audience version.
Stochastic processes and their applications, Berlin	Jul. 2009	Numerical methods for random spatial permutations	Abstract, slides.
Workshop on quantum spin systems and applications in quantum computation, Tucson	Jun. 2009	Lattice quadrupling for percolation in quantum networks	Slides.
UA Math Dept. comprehensive examination	Jan. 2009	Remarks on interacting spatial permutations and the Bose gas	Paper, slides.
UA Math Dept. Graduate Student Colloquium	Oct. 2008	Computational methods in percolation	Abstract, slides.
UA Math Dept. Applied Brown Bag	Oct. 2008	High performance arithmetic	Abstract, slides in PDF, slides in PowerPoint.
UA Math Dept. Math. Phys. Seminar	Oct. 2008	Lattice quadrupling for percolation in quantum networks	Abstract, slides.
UA Math Dept. Software Interest Group	Apr. 2008	C for math folks	Abstract, slides.
UA Math Dept. Math. Physics Seminar	Apr. 2008	Monte Carlo methods for interacting spatial permutations	Abstract, slides, paper.
UA Math Dept. Graduate Student Colloquium	Mar. 2008	Quantum mechanics for math grads	Abstract, paper, 1D SE solver, 2D SE solver, two-state example.
UA Math Dept. Graduate Probability Seminar	Feb. 2008	The Metropolis-Hastings algorithm by example	Abstract, paper, code.
UA Math Dept. Graduate Student Colloquium	Jan. 2008	Numerical Differential Geometry in Matlab	Abstract, slides, Matlab code
UA Math Dept. Graduate Student Colloquium	Sep. 2007	Is 2 a random number?	Abstract, paper
UA Math Dept. Graduate Student Colloquium	Feb. 2007	The Householder transformation, Swiss Army knife of numerical linear algebra	Abstract, paper
UA Math Dept. Software Interest Group	Oct. 2006	Perl for the working (or aspiring) mathematician	Abstract, slides
UA Math Dept. Graduate Student Colloquium	Sep. 2006	Tensorama	Abstract, paper (see the section titled Tensors)
UA Math Dept. Software Interest Group	Apr. 2006	Concrete abstract algebra in Python	Abstract, slides, paper
UA Math Dept. Graduate Colloquium	Mar. 2006	A walking tour through projective spaces	Abstract
UA Math Dept. Graduate Colloquium	Jan. 2006	Computation in finite fields	Abstract, paper
UA Math Dept. Software Interest Group	Sep. 2005	Introduction to the vim text editor	Slides
UA Math Dept. Graduate Colloquium	Sep. 2005	Linear feedback shift registers	Abstract, paper
Master’s thesis defense, ASU	Apr. 2005	Codes and Curves	Abstract, slides, thesis
ASU/Lockheed Martin Seminar Series	Oct. 2004	High performance arithmetic	Slides in PDF, slides in PowerPoint
ASU Graduate Student Seminar Series	Sep. 2004	An introduction to coding theory for mathematics students	Abstract, paper
ASU Graduate Student Seminar Series	Apr. 2004	Linear feedback shift registers	Abstract, paper
DesignCon, San Jose	Feb. 2004	A reference design for FPGA-based Linux applications
DesignCon, San Jose	Feb. 2004	A reference design for embedded ucLinux on a Motorola ColdFire processor
ASU Graduate Student Seminar Series	Dec. 2003	Computation in finite fields	Abstract, paper

Documents

Formal papers:

• Doctoral dissertation, March 2010: Critical behavior for the model of random spatial permutations. PDF format, PostScript format, DVI format.
  Frontmatter, Chapter 1: Scientific context, Chapter 2: The model of random spatial permutations, Chapter 3: Random variables, Chapter 4: Markov chain Monte Carlo methods, Chapter 5: The swap-only and swap-and-reverse algorithms, Chapter 6: Band updates, Chapter 7: The worm algorithm, Chapter 8: Delta H computations, Chapter 9: Algorithms for single MCMC runs, Chapter 10: Batching of MCMC runs, Chapter 11: Results, Chapter 12: Future work, Appendix A: Bose-gas derivation of random permutations, Appendix B: Error bars, autocorrelation, and batched means, Backmatter.
• Dissertation appendix: Notes on exponentially correlated stationary Markov processes.
• Preprint of proceedings note for the Center for Simulational Physics at the University of Georgia: A worm algorithm for random spatial permutations. The final publication is available in Volume 4, 2010, of Physics Procedia.
• Preprint of condensed version of dissertation, submitted to Journal of Statistical Physics (December 2009): Shift in critical temperature for random permutations with cycle weights. arXiv:0912.4292 [cond-mat.stat-mech]; local copy. The final publication is available at www.springerlink.com.
• French language examination: original paper, my translation.
• Written comprehensive examination paper: Remarks on interacting spatial permutations and the Bose gas (January 2009).
• Master’s thesis (May 2005): Curves and Codes.

Expository mathematical papers :

• Is 2 a random number?
• The Metropolis-Hastings algorithm by example
• The Householder transformation, Swiss Army knife of numerical linear algebra
• Computation in finite fields
• Concrete abstract algebra in Python
• Quantum mechanics for math grads
• An introduction to coding theory for mathematics students

Course notes:

• Probability (Math 564, spring 2007).
• Probability and statistics (Math 564 spring 2007 and Math 567A spring 2008): under construction.
• Geometry/topology (Math 534A-B, 2005-2006).

Notes for qualifier preparation (all perpetually under construction):

• Geometry/topology qual solutions.
• Complex analysis.
• Real analysis.
• Abstract algebra.

Course/project papers:

• Project on Berlekamp’s algorithm for incoming graduate students, fall 2009 integration workshop.
• Some problems in stochastic differential equations (Math 565C, spring 2008): problems and figures. Source code: Brownian motion and bridges with statistical verification, textual output, and graphical output; 2D bridges with graphical output. Damped Brownian motion, Brownian motion on an ellipse, noisy exponential growth, noisy exponential decay, Brownian bridges one way, Brownian bridges another way, complex Brownian motion, exponential growth with multiple noise sources, first martingale problem, second martingale problem, mean-reverting Ornstein-Uhlenbeck process, stochastically forced vibrating string, Feynman-Kac formula.
• Final project for Theoretical Statistics (Math 567A, spring 2008): paper, source code.
• Fall 2006 RTG paper: The Discrete Berezin Integral.
• Poisson summation and the discrete Fourier transform (spring 2006): Math 523B term paper, slides.
• The DFT and the FFT.
• Some examples of Fourier approximation, with nice plots.

Elementary write-ups:

• Getting hit by lightning ... in the long run.
• Choice of base in exponential models.
• Mathematical induction.
• The cubic formula.
• Derivation of the sum and differences formulas for sine and cosine.
• The gamma integral formula.
• How to manually compute a square root.
• What’s the difference between probability and statistics?
• An argument-settling (?) approach to the Monty Hall problem

HTML:

• Notes for SWIG talk on Web-page design.
• Template home-page file (all in one file), instructions.
• Second template home-page file (uses multiple files), one file you need, another file you need.
• HTML quick reference, brief introduction to HTML.

LaTeX:

• Template PhD dissertation, tar file.
• Template file for using figures in LaTeX, tar file.
• Template Beamer presentation with included EPS figure and PDF as output: PDF, main .tex file, auxiliary .tex file, DVI build info, PDF build info, tar file.
• Template paper with abstract, table of contents, appendices, bibliography, and index, with hyperlinked PDF as output: PDF, tar file. The build script requires kmkidx, which you should download and put in your path.
• Template quiz file, LaTeX source.
• A blank template homework file: template.tex, template.pdf.
• Calculus I exams from spring 2007: Instructions, tar file.
• A quick how-to for typing up homework in LaTeX: sample.pdf. Source files: sample.tex, isomath.tex, mathenv.tex (download all three).

Typesetting misc.:

• A table of some special characters in Windows.

Software misc.:

• How to use the Unix command-line mail tool (and you thought Pine was old-fashioned ...)
• A simple introduction to the make utility
• A table of the C language’s operator precedence
• How to do a sprintf in Python: sprintf.html.
• A reminder for various shell-script syntax
• A short course in linking and loading

Rubik’s cube:

• How to solve the Rubik’s cube (Nourse method), one-face-moves, threefold moves, same, with more verbage.

Miscellaneous (pre-grad school):

• DDA scaling and shifting formulae.

Software

Github repositories: https://github.com/johnkerl

Note: all current content is in Github. Items below are snapshots with few or no updates since 2010.

PHP:

• Collapsible trees, for a previous version of this homepage; source code here.
• Small-integer factorizer with AJAX+PHP, same but with form+PHP.
• A frivolous cycle decomposer; source code here.

Python:

• pgr, a plotter which sends tabular file data to pylab: documentation, main script, required I/O module.
• Matrix arithmetic (using the matrix module sackmat_m.py): Matrix multiply, matrix add, matrix subtract, matrix inverse, matrix power, determinant, trace, real matrix exponential, complex matrix exponential, real matrix square root (Denman-Beavers), complex matrix square root, real matrix A^*A, complex matrix A^*A, row reduction, row echelon form, row rank, kernel basis, kernel basis, Gram-Schmidt orthonormalization, Jacobi eigensolver for real symmetric matrices, tester for the Jacobi eigensolver, Householder-matrix generator, another Householder-matrix generator, another Householder-matrix generator, rotation-matrix generator, QR decomposition, real polar decomposition, complex polar decomposition.
• Simple MCMC (Metropolis-Hastings) sampling for the 2D Ising model.
• Misc. matrix generators: oneeye, tridiag, discrete Laplacian with Dirichlet boundary conditions, discrete Laplacian with periodic boundary conditions.
• Numerical calculus: 1-D numerical differentiation, 1-D numerical integration, contour integrals, another contour integral, streamlined complex integration.
• Taylor polynomials, Lagrange interpolation module, program.
• Real and complex I/O.
• Fourier processing: Fourier-series approximation, Fast Fourier Transform module and program. (Uses pyrcio_m.py.)
• Probability: some Markov chain hackwork, 1D and 2D Brownian motion and Brownian bridge, beta-distribution PDF, beta-distribution PDF plotter. The latter uses Tom Loredo’s special functions.
• Gaussian normal routines normalpdf, normalcdf, invnorm, erf, erfc, and inverf: normal_m.py.
• Real/complex abstraction layer for conj(), real(), imag(), phz(): cplxreal_m.py.
• Random-scalar-complex generator: module, tester.
• Real random-matrix generator (m × n, GL(n), SL(n), O(n), SO(n), symmetric, skew-symmetric): program randmat, module randmat_m.py.
• Complex random-matrix generator (m × n IID mean-square-1, GUE, IID mean-square-1 XX^t): program randmatc, module randmatc_m.py.
• Statistics: module, 1D histogram, 2D histogram, mean, standard deviation, pair covariance, pair correlation, covariance matrix, correlation matrix, univariate linear regression, noisy-line generator, noisy-sine generator, stratified random sampling.
• Sliding-window data smoother: program, module.
• SACK: A simple algebra calculator. Includes some elementary group routines, some linear-algebra functionality, and some support for computation in real tensor algebras. Documentation, source tree, tar file.
• Routines for printing images, cycle decompositions, and orders of moves on the Rubik’s cube: module, demo, demo output.
• A utility module: kerlutil.py; float to hex: float2hex_m.py; how to do a sprintf in Python: sprintf.html.
• Tar file of Python programs: python.tgz.

Java:

• Brownian motion and Brownian bridges: Brownian motion, test routine, Brownian bridges, test routine, I/O module, statistics module.

Perl:

• Makefile generator (“yet another make-make”): yamm, sample program 1 for yamm. More programs using yamm are in the tar file gen.tgz.
• Substitute for makeindex which correctly handles alphabetization of text including \mathbb etc. in LaTeX files: kmkidx.
• Tools for formatting and manipulating tabular data in plain ASCII: colprint, mwxprint, parside, interleave, creach, twoper, xpose, cpick, dashes, caytex, mattex.
• Tools for large-scale search and replace: krepl, mrepl. Variants of familiar tools: bffind, mgrep.
• Windows workalikes for familiar Unix tools: kbc, grep.pl, head.pl, tail.pl, uniq.pl, wc.pl, whereis.pl, My_win_glob.pm, w.pl, notes about Perl and Windows file associations, toc.pl which works around the Windows association/redirection bug.
• Unix workalikes for familiar Windows tools: grepr, lsr, rmr.
• fixrlinks: When I rsync -rtlz between hosts on which the spelling of my home directory differs (e.g. /u5/kerl on one and /home/kerl on the other), rsync doesn’t change the symlink referents. This script can be run after an rsync to fix that.
• finddang: Reports on dangling symlinks.
• Miscellaneous text manipulation: upcase, downcase, scrunch, printtwice, printtwicej, printthricej, chop, dspc, dspcw, linesort, detab, retab, fdiff, nocolon, wordfreq.
• Numerical calculus: series estimation, another series estimation, 1-D numerical differentiation, 1-D numerical integration, 2-D numerical integration, path integral one way, same path integral another way.
• Floating-point matrix arithmetic: Matrix multiply, matrix add, matrix inverse, determinant, row reduction, row echelon form, row rank, kernel basis, Gram-Schmidt orthonormalization, Jacobi eigensolver for real symmetric matrices, elementwise vector sum, elementwise vector difference, elementwise vector product, vector times scalar, dot product, vector-to-scalar sum, vector-to-scalar sum of squares, matrix library file PMATLIB.pm.
• Generators of special-purpose matrices: Hilbert matrix (famously near-singular), identity, DDA A, DDA Q.
• Fast Fourier transform and associated tools: radix-2 FFT, complex rectangular-to-polar, complex polar-to-rectangular, power sum (for discrete Parseval).
• Data generators for testing FFT implementations: pcisgen, pspikegen, phatgen, prampgen, prandgen.
• Misc. numerical routines: Lagrange interpolation, partial sums, binomial coefficients and Pascal’s triangle, extended GCD using Blankinship’s algorithm.
• Tar file of Perl programs: perl.tgz.

C/C++:

• C project for enumerating constrained (e.g. self-avoiding) walks on the plane integer lattice: README file, source code.
• C/bash project for bond percolation on the two-dimensional square lattice (used for my PhD computational exam at UA): PDF with mathematical background, source code.
• SPFFL, a small-prime finite-field C++ library (used for my master’s thesis): documentation, source code.
• Misc. general-purpose C tools: source code.
• Some simple X11 C++ programs: xgr documentation, xfim documentation, tree, tar file.
• Examples of using function pointers in C: without typedefs, with typedefs.

LISP:

• Cycle decomposition etc. for the Rubik's cube.

Fortran:

• Simple programs: hello in F77, hello in F90, F90 I/O examples, how to find machine epsilons.
• Some library routines: Simpson adaptive quadrature, LU decomposition, matrix arithmetic, matrix I/O.
• Some library routines: Crout, back-substitution,
• Drivers: matrix arithmetic, single-precision Simpson, double-precision Simpson, LU decomposition, LU solver.
• Makefile.
• Tar file.

Matlab:

• Vector calculus / differential geometry: An illustration of the Jacobian, tangent planes to S2 using graph and spherical coordinates.
• Final project for Math 537A (Riemannian geometry): gdgproj.m, lecture notes, tar file of complete directory. (You need gdgproj.m and all the other *.m files.)
• Binary I/O: fbinread, fbinwrite.
• Plot animation: param_movie.m.
• Misc.: cmpvecs, vecrange.

Research

• Dissertation topic: Critical behavior for the model of random spatial permutations.
  • Research statement, lay version.
  • Doctoral dissertation: Critical behavior for the model of random spatial permutations. PDF format, PostScript format, DVI format; defense slides.
    Frontmatter,
    Chapter 1: Scientific context,
    Chapter 2: The model of random spatial permutations,
    Chapter 3: Random variables,
    Chapter 4: Markov chain Monte Carlo methods,
    Chapter 5: The swap-only and swap-and-reverse algorithms,
    Chapter 6: Band updates,
    Chapter 7: The worm algorithm,
    Chapter 8: Delta H computations,
    Chapter 9: Algorithms for single MCMC runs,
    Chapter 10: Batching of MCMC runs,
    Chapter 11: Results,
    Chapter 12: Future work,
    Appendix A: Bose-gas derivation of random permutations,
    Appendix B: Error bars, autocorrelation, and batched means,
    Backmatter.
  • Dissertation appendix: Notes on exponentially correlated stationary Markov processes.
  • Slides for a job talk: half on general MCMC methods, half on my dissertation research.
  • Slides for 15-minute talk at the February 2010 Workshop of the Center for Simulational Physics at the University of Georgia; manuscript for same.
  • Slides for 15-minute talk at the January 2010 joint meetings.
  • Preprint of condensed version of dissertation, submitted to Journal of Statistical Physics: arXiv:0912.4292 [cond-mat.stat-mech]; local copy.
  • VIGRE proposal for spring 2010; statement of merit and impact; VIGRE report for spring 2010.
  • September 2009 talk for the UA Mathematical Physics Seminar: abstract, slides.
  • July 2009 talk for the Berlin SPA: abstract, slides.
  • VIGRE proposal for summer/fall 2009; VIGRE report for summer 2009, VIGRE report for fall 2009.
  • Written comprehensive examination paper; slides for oral comprehensive examination (January 2009).
  • Pseudocode for MCMC simulation of the N2 model (spring 2008).
  • VIGRE proposal for fall 2008.
  • April 2008 talk for the UA Mathematical Physics Seminar: abstract, slides, paper.
• Spring 2008 (follow-up in summer 2009): Independent study under Jan Wehr on a percolation problem arising in quantum networks.
  • Jun. 2009 lecture slides.
  • Oct. 2008 lecture slides.
• Fall 2006 Research Tutorial Group.
  • Paper on the discrete Berezin integral.
• Master’s thesis, May 2005:
  • Thesis: Curves and Codes.
  • Defense slides.

Teaching

Courses taught:

• Fall 2008, Math 124 (calculus I with applications), section 8:
  • Calendar and homework list, my weekly schedule.
  • Course policy.
  • Links: WebAssign, D2L, math.arizona.edu/~calc.
  • Study guide for the preliminary exam: prep.math.lsa.umich.edu/pmc.
  • Exam 1 study guide, exam 1, exam 1 solutions.
  • Exam 2 study guide, exam 2, exam 2 solutions.
  • Exam 3 study guide, exam 3, exam 3 solutions.
  • Exam 4 study guide, exam 4, exam 4 solutions.
  • Worksheet on optimization and related rates.
  • Worksheet on l’Hôpital’s rule.
• Spring 2007, Math 124 (calculus I with applications), section 8:
  • Course policy, calendar and homework list, math.arizona.edu/~calc.
  • My weekly schedule.
  • Exam 1 study guide, exam 1, exam 1 solutions.
  • Exam 2 study guide, exam 2, exam 2 solutions.
  • Exam 3 study guide, exam 3, exam 3 solutions.
  • Exam 4 study guide, exam 4, exam 4 solutions.
  • Worksheet on optimization and related rates.
  • Information about the final exam.
• Fall 2006, Math 111 (trigonometry), section 7:
  • Course policy addendum, calendar and homework list.
  • My weekly schedule.
  • Exam 2 study guide, exam 2, exam 2 solutions.
  • Course page: math.arizona.edu/~trig.
• Fall 2006 and spring 2007, Math 511A-B (abstract algebra), super TA:
  • Rob Pawloski’s problem sets.
  • Savitt’s assignments.
• Spring 2006, Math 110 (college algebra):
  • Course policy addendum, calendar and homework list.
  • My weekly schedule.
  • Exam 4.
• Fall 2005, Math 110 (college algebra):
  • Course policy addendum, calendar and homework list.
  • My weekly schedule.
  • Exam 4, exam 4 solutions.

Information for students:

• Documents related to freshman math:
  • Why is the sum-of-consecutive-squares formula true?
  • A brief derivation of the sum and differences formulas for sine and cosine.
  • The gamma integral formula.
  • What’s with the base in exponential models?
  • We have a quadratic formula. Have you wondered if there’s such a thing as a cubic formula? In fact, there is.
  • Why cross the z’s?
• Links: math department home, math department tutoring, math department syllabus information, UA academic calendar, UA General Catalog, University Learning Center, Disability Resource Center.

Notes from the professional-development course, 2005-06 (first teaching year):

• (1) Day one
• (2) Lecture notes
• (3) Organization and presentation
• (4) Classroom management, rapport, and motivation
• (5) Observations of other instructors
• (6) The Rule of Four
• (7) Policies
• (8) Feedback for the professional-development course
• (10) Writing exams
• (11) A lesson plan on word problems
• (12) Grading
• (13) Observations of other instructors
• (14) Semester two compared to semester one

Other:

• Teaching statement
• The exemplar model in mathematics instruction

Personal

• Hobbies:
  • Photos
  • Woodworking
  • Brevity
• Misc. links:
  • People I have been
  • People and things which I am not
  • My heroes in the media
  • Unit-quaternionic description of myself: jk=i. :)

About this site

This is a collection of papers, presentation materials, and software: mostly mathematical, and mostly from my 2005-2010 PhD years in the University of Arizona Department of Mathematics ... although some new content does trickle in from time to time. :)

Image credit for the clock layer of the background image is here. The formula/code/graph layers are my own, from various parts of this site.