Ordinary polytopes are known as a non-simplicial generalization of the cyclic polytopes. The face vectors of ordinary polytopes are shown to be log-concave.

No abstract

K-12 math education is struggling as despite obvious job market benefits, several students choose to discontinue math education when given the possibility. At the same time, advances in learning technologies now enable modes of learning that were impossible up until ten years ago. In this study we analyse math education from scratch by returning to the hunter-gatherer time period and empirically observing how mathematics is present the lives of a San tribe in Southern Namibia with ethnographic analysis. With this work, we propose two high level design considerations for integrating math learning technologies with the hunter-gatherer way of living: (1) integration of learning technologies and real world objects; and (2) the introduction of physical activity and social communication to math education. We discuss how these two design considerations could boost students' motivation to continue learning math also beyond the formal school environment.

The Cayley graph construction provides a natural grid structure on a finite vector space over a field of prime or prime square cardinality, where the characteristic is congruent to 3 modulo 4, in addition to the quadratic residue tournament structure on the prime subfield. Distance from the null vector in the grid graph defines a Manhattan norm. The Hermitian inner product on these spaces over finite fields behaves in some respects similarly to the real and complex case. An analogue of the Cauchy-Schwarz inequality is valid with respect to the Manhattan norm. With respect to the non-transitive order provided by the quadratic residue tournament, an analogue of the Cauchy-Schwarz inequality holds in arbitrarily large neighborhoods of the null vector, when the characteristic is an appropriate large prime.

Convex or concave sequences of n positive terms, viewed as vectors in n-space, constitute convex cones with 2n − 2 and n extreme rays, respectively. Explicit description is given of vectors spanning these extreme rays, as well as of non-singular linear transformations between the positive orthant and the simplicial cones formed by the positive concave vectors. The simplicial cones of monotone convex and concave vectors can be described similarly.