Research

Most of my mathematical research is not available in electronic form. Consult my curriculum vitae for references. Below are links to work currently available in electronic form.

The Ehrenfeucht-Mycielski sequence.

This is a psuedo-random sequence of binary digits defined by a requirement of non-repetition.

• The Fermat Roberval Inequalities. An expository historical commentary and the 17th century solution of the quadrature problem for power curves.

• The Derivative as Slope of Tangent Line. We examine critically the statement that the derivatives measures slopes of tangent lines. (Easy mathematical reading.)

• A Geometric Approach to Calculus. Calculus without the limit of difference quotients. This is a sequel to "The Derivative as Slope of Tangent Line".

• Remarks on Decoupling Inequalities for Random Multi-linear Forms Possibly contains some new proofs or, at any rate, more explicit constants. One hates to say anything is new, because it almost never seems to be true.

• Slant Asymptotes and the Legendre Transform A convient calculus-based method for finding asymptotes to curves.

• Repairing the Discontinuous Function A note on redefining functions so as to make them continuous or Riemann integrable.

• Extensions of the Riemann Integral This is a follow-up on Repairing the Discontinuous Function. It shows that the domain of the Riemann integral can be extended, with almost no effort, to include many more functions, such as the characteristic function of the irrationals (and even many non-measurable functions), without losing any of the cherished properties of the Riemann integral.

• Note on Segmented Series Various examples of conditionally convergent series constructed using a slight extension of some recent work of Schramm, Troutman, and Waterman on generalizations of the alternating series theorem.

• A mostly expository paper on orthogonal regression, a variant in which one seeks to minimize the sum of squared distances to a test line rather than the sum of squared vertical distances.

• A paper that considers the Birthday Problem when birthdays may be dependent. IMPORTANT: please use the following revised version . The original needed correcting, but I still own it, so it should remain.

• Followup paper on the Birthday Problem when birthdays are dependent. Contains a number of further examples of affiliated random variables and further explores the relationship between affiliation and association.

• The expected time to find a string in a random binary sequence, a mainly expository paper which surveys known results on the subject of the title.

• An Inequality Related to the Birthday Problem, , a short note on the subject of the title. Maple command file referenced in the paper.

• DeBruijn Strings, Double Helices, and the Ehrenfeucht-Mycielski Mechanism a paper that explores connections between the Ehrenfeucht-Mycielski sequence and DeBruijn sequences.

• My rediscovery of Stewart's Theorem. The proof may be of interest as an alternative to the usual one involving the law of cosines. Also, the kinetic energy decompositon may be of some slight interest.