Joel Zeitlin

NOTES ON D-OPTIMAL DESIGNS

Abstract The purpose of this paper is to exhibit new infinite families of D-optimal (0, 1)-matrices. We show that Hadamard designs lead to D-optimal matrices of size ( j , mj ) and ( j − 1, mj ), for certain integers j ≡ 3 (mod 4) and all positive integers m . For j a power of a prime and j ≡ 1 (mod 4), supplementary difference sets lead to D-optimal matrices of size ( j , 2mj ) and ( j − 1, 2mj ), for all positive integers m . We also show that for a given j and d sufficiently large, about half of the entries in each column of a D-optimal matrix are ones. This leads to a new relationship between D-optimality for (0, 1)-matrices and for (±1)-matrices. Known results about D-optimal (±1)-matrices are then used to obtain new D-optimal (0, 1)-matrices.

D-optimal weighing designs for four and five objects

For j = 4 and j = 5 and all d j, the maximum value of detXX , where X runs through all j d (0,1)-matrices, is determined along with a matrix X0 for which the maximum determinant is attained. In the theory of statistical designs, X0 is called a D-optimal design matrix. Design matrices that were previously thought to be D-optimal, are shown here to be D-optimal.

D-optimal weighing designs for six objects

Abstract. For all integers m≥6, we determine the maximum value of det XTX, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix XTX is discussed.

Maximalj-Simplices in the Reald-Dimensional Unit Cube

Department of Mathematics, California State University Northridge,Northridge, California 91330Communicated by the Managing EditorsReceived July 27, 1996For each positive even integer j there is an infinite arithmetic sequence of dimen-sions d for which we construct a j-simplex of maximum volume in the d-dimen-sional unit cube. For fixed d, all of these maximal j-simplices have the same Grammatrix, which is a multiple of I+J. For j even, a new upper bound for the volumeof a j-simplex in the d-dimensional unit cube is given.

Apportionment and the 2000 Election

Michael Neubauer (michael.neubauer@csun.edu) received his Ph.D. from the University of Southern California under the direction of Robert Guralnick in 1989 and is now at California State University, Northridge. He became interested in apportionment theory while teaching the topic and after many conversations on the topic in between swimming laps he and Joel Zeitlin finally got the idea for this article. He very much enjoys the company of his wife and his two kids.

SMALL STEPS TO A STUDENT-CENTERED CLASSROOM

ABSTRACT Research supports the view that students should play an active role in the mathematics classroom. This paper describes several techniques that have been used in a variety of mathematics classes (remedial, pre-calculus, liberal arts mathematics, calculus, and geometry) to shift the mathematics classrooms experience from an instructor-centered one to a student-centered one.