Teaching

Usually Useful Pamphlets

• The Checklist (pdf, 3 pages) has important ingredients of good mathematical writing. It also has some of the abbrevs that I use in grading. See also Studying (pdf).

• Here is my general terminology (pdf). If you don't know it already, it is crucial that you learn the Math-Greek alphabet (pdf), which shows how I use it in class, and on pamphlets; it also has my special symbols, e.g, the Golden ratio, or the Riemann zeta fnc.

  Three authors at UC Davis wrote Some Common Mathematical Symbols and Abbreviations (with History) (pdf) which has a nice list of symbols, with the presumed introducer of each. I don't use all of these symbols/phrases in class. A very few of them, I use slightly differently from how the above link shows.

• For email to each other, here are mathematical conventions for the QWERTY keyboard, as well as a partial list of mathematical notation for Maple.
• Have you ever wondered: Does Zero = One? (pdf). (Has: Calculus Poofs, Poofs by Induction, Combinatorial Poofs, even Poofs from Complex Analysis.)
• Two pages on Converting eventually-periodic numerals to fractions (pdf).

Puzzles

• The Omniscient Alien.

Geometry stuff

• Two proofs of the Euler-line thm (pdf).
• Triangles and other convex shapes (pdf).

Calculus-ish stuff

• Proofs of: Intermediate value Theorem (pdf) and Addition and Multiplication are continuous (pdf).
• Proof of the Mean Value Thm and L'Hopital's Rule (pdf).
• Vector Fields (pdf).
• Algorithms for solving some differential equations (pdf).
• The Infinite Snowplow Pile-up (pdf).
• FTODE: Existence and uniqueness of solution to an ODE (pdf).
• The Laplace transform (pdf) has an example, a simplification of tapping on a bell.
• Table of Laplace transforms (pdf).
• A C-infinity function which is not everywhere analytic (pdf).
• One way to compute the curvature of an ellipse (pdf).
• Planetary orbit (pdf) computations.
• Volume of the n-ball (pdf).
• A 2-constraint Lagrange multiplier (pdf) example.

• Pamphlet Differentiating a bilinear function (pdf) shows how to generalize the Product Rule to other bilinear functions.

  Least Squares and matrices (pdf): Derivation of a formula for Least-squares fitting of a line to data points (also has HW problems). Available are two examples of a more-general curve-fitting formula (txt).

• There does not exist a real-valued f continuous exactly on Q (pdf).
• Rings, ordered-fields, Archimedean fields, and complete ordered-fields (pdf). [Not quite finished]

Number theory stuff

• The Euclidean algorithm can be presented in table-form; I call this form the Lightning-bolt algorithm (pdf), because the update-rule looks like a lightning-bolt (used thrice). Here is a practice sheet for LBolt (pdf).

  The first page of Algorithms in Number Theory (pdf), uses LBolt iteratively to compute the GCD of a list of integers, together with its list of Bézout multipliers. Page 2 uses LBolt to solve linear congruences: Find all x where 33x is mod-114 congruent to 18.

  Here are examples of fusing congruences (txt) using LBolt

  Everybody loves the Euler-Fermat thm. Available is Using EFT to solve 102⁷⁰ + 1 =113= b³⁷ (txt), from Prof. William Stein's book.

• A proof of the Chinese Remainder Theorem (pdf) [CRT] with many additional details. Proves that Euler phi is a multiplicative fnc. Here are some An example of using CRT to count roots of a polynomial.
• Partial Theorem List (pdf) for Number Theory.
• Special case of Dirichlet's Thm (pdf), showing that there are infinitely many primes in -1+6Z. Also, for SeLo, comments on the exposition.
• Pythagorean Triples (pdf).
• Mersenne primes and Even Perfect numbers and Congruences in Number Theory (pdf).
• Elementary facts about Fibonacci Sequences (pdf). The Fibonacci Speedup (txt) shows how to use repeated squaring and modular arithmetic to rapidly compute the last two digits of the 1024^th Fibonacci number.
• Intro to Multiplicative Functions (pdf) and convolution of number-theoretic functions, such as Euler-phi. Application of Generating functions and Mobius inversion (pdf) to counting irreducible polynomials over a finite-field.
• Proof of Quadratic Reciprocity (pdf). In particular, proofs of the Eisenstein and Gauss Lemmata.
• Finite fields have cyclic multiplicative groups (pdf). This also proves the Primitive Root Thm, and Korselt's Thm characterizing the Carmichael numbers.
• Hensel's Lemma (pdf), for a lifting a mod-p root of a polynomial, to be a root mod pⁿ.
• Notes-in-progress on Fermat's SOTS (Sum-Of-Two-Squares) thm.
  Slides showing how the Euclidean Algorithm writes a given N as a Sum Of Two Squares (pdf). Such an N is a SOTS in the notes.
• A characterization of the irreducibles elements in the ring of Gaussian integers (pdf).
• Liouville's Theorem (pdf) on approximating reals by rationals. Also: Hilbert's proof that e is transcendental (pdf).
• Proofs of Bertrand's postulate (pdf) and Chebyshev's thm, and other results on the density of prime numbers. See also Binomial Coefficients (pdf).
• Smith Normal Form and Integer solutions to linear equations (pdf).
• I janned up some emacs-lisp code for the Miller-Rabin probabilistic primality test (txt).

• PDF images of the elliptic surface     y²  =  x³ -2x + 1.

  These images were made with Gnuplot: ell-curve-cntr4 and ell-curve-cntr2 and ell-curve-cntr1 and ell-curve-cntr0.5.pdf. Here is a singular curve (the light purple-reddish self-intersecting curve) on the same surface.

Combinatorical stuff

• A std proof of the Inclusion-Exclusion principle (pdf).
• An intro to the Chromatic polynomial of a graph (pdf).
• Notes on bipartite graphs (pdf).
• Generating-Function notes (pdf).
• Notes on Ramey-like theorems (pdf).

Algebrish stuff

• Primer on Polynomials (pdf).
• Basic Algebra definitions (pdf). [Currently this has some group-theory results, which need to be moved elsewhere.]
• Solutions to problems in Gallian's text (pdf).
• Example of Burnside's Lemma, for coloring the cube (pdf). [This could also be filed under Combinatorics.]
• Algorithmic exposition of Fund. Thm of Symmetric Polynomials (pdf).
• Voila! a proof of the Eisenstein Irreducibility Criterion (pdf).
• Proof of Fund. Thm of Finite Abelian groups (pdf).
• Here we explore N×N determinants (txt) and Cramer's “Rule”, i.e Theorem (txt).
• A proof of the Rank-Nullity Theorem (pdf), and also a proof that RREF is unique (pdf), stated appropriately.
• Nullspaces of commuting transformations (pdf); applies Linear Algebra to constant-coeff linear differential equations.
• An example of the Gram-Schmidt orthogonalization algorithm. Also, Rotation matrices in 3 dimensions (txt); the general orthogonal matrix.

• Linear Recurrence using matrices (pdf) solves a fibonacci-like recurrence; it makes a cryptic reference to Jordan Canonical Form.

  Its method is to attempt to diagonalize a matrix.

  Examples: a 2x2 diagonalizable matrix. And a 3x3 matrix with only 2 dimenions of eigenvalues. Also powers of a diagonalizable 4x4 matrix.

  Finally, here are examples showing Trace and char-poly are preserved under conjugation.

• The Triangular Matrix Lemma [just page 4 in JCF] (pdf), and the Cayley-Hamilton Theorem (pdf).
• A glance at projections (pdf), and the dictionary game.
• That vert-and-hor shears generate SL₂ (pdf).
• Distance between flats (pdf), which will eventually compute the distance between two arbitrary flats (translated subspaces).