Tiling the integers with translates of one finite set

A set is said to tile the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power size, it was solved by D. Newman [J. Number Theory 9 (1977), 107--111]. We solve it for sets of size having at most two prime factors. The conditions are always sufficient, but it is unknown whether they are necessary for all finite sets.Comment: 12 page

Profiles of Promise: Students With Learning Difference Prepare For Optimal Foreign Language Learning

Many secondary school counselors discourage students with diagnosed native language deficits from foreign language studies. They hold the unsubstantiated belief that these studies would jeopardize their hard-earned gains in native language learning. Research disproves that position, and some writers advocate foreign language study to improve language learning and enhance cognitive strategies. Improvement occurs more readily when the instructor has the expertise to choose from a repertoire of teaching styles the one most compatible with the learning style of the student. With the full inclusion promised by the regular education initiative becoming a reality, this study explores research, design, and team-teaching a language enrichment course to prepare these “at-risk” students for optimal foreign language learning. This qualitative study records interprets and analyzes the impact of the language enrichment course, known as HOLA!, on two selected students. The students\u27 academic histories were reviewed, and their class experiences were recorded. Student progress was assessed after the course. Students were interviewed as were their parents and their teachers regarding their attitudes toward the program. Student self-images, their awareness of their learning styles, and what they learned about the study of language were assessed. Analysis indicates that students did benefit from HOLA! in different ways, but the benefits in common were two-fold: an awareness of the strategies needed to succeed in the foreign language class and the recognition of their need to be self-advocates in pursuit of optimal learning. Judging by their excellent grade reports, these abilities transferred to other courses

On the Ergodic Properties of Certain Additive Cellular Automata over ZmZ_{m}

In this paper, we investigate some ergodic properties of $Z^{2}$-actions $T_{p,n}$ generated by an additive cellular automata and shift acting on the space of all doubly -infinitive sequences taking values in $Z_{m}$.Comment: 5 pag

Who Should be Entitled to Claim the New Education Credits?

Professor Coven believes that the education assistance provisions enacted in 1997, while long overdue, were ill-considered and poorly constructed. Focusing on what should have been the simple question of who is entitled to claim an education tax credit, this report illustrates the harm that inadequate drafting produces. While section 25A appeared to deny a credit to a dependent child, an unfair and unwise result, the proposed regulations allow parents to shift the credit to their children but only on the forfeiture of the deduction for the personal exemption. That rule, says Coven, imposes a harsh penalty on low and middle-income taxpayers but no penalty at all on high-income taxpayers. Neither result is consistent with the overall aim of the education credits. This report is an excerpt from a broader study of the drafting deficiencies in the educational assistance provisions currently being undertaken by the author. Because of the poor quality of these sections, considerable corrective legislation will be required to fulfill the promise of the 1997 legislation