Motivated by the Erdős multiplication table problem we study the following question: Given numbers N1, . . . , Nk+1, how many distinct products of the form n1 · · ·nk+1 with 1 ≤ ni ≤ Ni for i ∈ {1, . . . , k + 1} are there? Call Ak+1(N1, . . . , Nk+1) the quantity in question. Ford established the order of magnitude of A2(N1, N2) and the author of Ak+1(N, . . . , N) for all k ≥ 2. In the presen...

Abstract This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc. 136 (3) (2004), 513–524]. In particular, preservation word-hyperbolicity under taking free products is considered. Under mi...

This paper reports on strategies 26 Year 3 students used to solve a range of division word problems in a oneto-one interview, following the participation of half of the group in a teaching experiment. The focus here is on the strategies used by case study students to solve equivalent groups and times as many division tasks. Results suggest that young children are capable of solving complex divi...

FPGAs are an attractive platform for elliptic curve cryptography hardware. Since field multiplication is the most critical operation in elliptic curve cryptography, we have studied how efficient several field multipliers can be mapped to lookup table based FPGAs. Furthermore we have compared different curve coordinate representations with respect to the number of required field operations, and ...

A Technical Paper Submitted to IPPS 98 y Abstract Publications concerning parallel implementation of matrix-matrix multiplication continue to appear with some regularity. It may seem odd that an algorithm that can be expressed as one statement and three nested loops deserves this much attention. This paper provides some insights as to why this problem is complex: Practical algorithms that use m...

(June 7, 1998.) 0 This is the text of an article that I wrote and disseminated in September 1981, except that I've updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a simplified exposition of Deligne's extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The...

The present study explores the effect of two instructional methods for children with different levels mathematical skills. One these uses a conventional approach to learning multiplication and emphasizes memorization all arithmetic facts, whereas other method is based on psychological principles combines: a) small subset problems aided by color cues portable time-table, b) use single-step rules...

(June 7, 1998.) This is the text of an article that I wrote and disseminated in September 1981, except that I’ve updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a simplified exposition of Deligne’s extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The a...