Rick Norwood

The worm problem of Leo Moser

One of Leo Mosers geometry problems is referred to as the Worm Problem [10]: “What is the (convex) region of smallest area which will accommodate (or cover) every planar arc of length 1?” For example, it is easy to show that the circular disk with diameter 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539. Here we show that a solution to the Worm Problem of Moser is a region with area less than 0.27524.

An Improved Upper Bound for Leo Moser's Worm Problem

Abstract. A wormω is a continuous rectifiable arc of unit length in the Cartesian plane. Let W denote the class of all worms. A planar region C is called a cover for W if it contains a copy of every worm in W . That is, C will cover or contain any member ω of W after an appropriate translation and/ or rotation of ω is completed (no reflections). The open problem of determining a cover C of smallest area is attributed to Leo Moser [7], [8]. This paper reduces the smallest known upper bound for this area from 0.275237 [10] to 0.260437.

The maximum number of 2 by 2 odd submatrices in (0,1)-matrices

Let A be an m×n, (0, 1)-matrix. A submatrix of A is odd if the sum of its entries is an odd integer and even otherwise. The maximum number of 2×2 odd submatrices in a (0, 1)-matrix is related to the existence of Hadamard matrices and bounds on Turan numbers. Pinelis [On the minimal number of even submatrices of 0-1 matrices, Designs, Codes and Cryptography, 9:85–93, 1994] exhibits an asymptotic formula for the minimum possible number of p × q even submatrices of an m × n (0, 1)-matrix. Assuming the Hadamard conjecture, specific techniques are provided on how to assign the 0’s and 1’s, in order to yield the maximum number of 2 × 2 odd submatrices in an m × n (0, 1)-matrix. Moreover, formulas are determined that yield the exact maximum counts with one exception, in which case upper and lower bounds are given. These results extend and refine those of Pinelis.

TURNING DOUBLE-TORUS LINKS INSIDE OUT

The notation of t/i numbers is used to described knots and links on the double torus in two different ways, as a step toward the eventual classification of double-torus links.