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Abstract

It is well known that every rational integer has a finite or periodic p-adic expansion. In this paper a more general notion of p-adic expansion is introduced for algebraic integers, where given a number field K and a principal prime ideal p in K, a different choice of generator for p is allowed in each stage of the expansion. With the notion of p-adic expansion, we prove that there is always a finite or periodic p-adic expansion for every algebraic integer. Moreover, we prove a bound on the periodicity of the p-adic expansion that depends only on the number field K and the prime ideal p. The proof yields an algorithm for constructing such a p-adic expansion for elements in the ring O of algebraic integers of K, through finding an approximation to the closest vector on the lattice spanned by the unit group of O.As a special case we prove that, similar to rational integers, Gaussian integers are finite or periodic not only in p-adic expansion but also in π-adic expansion, where a fixed generator π for p is used in each stage of the expansion. Moreover, the time complexity of finding a π-adic expansion for a Gaussian integer is polynomial in the length of input, the period, and p, where p is the rational prime contained in p. We implement the algorithm for some quadratic number fields and provide examples which illustrate that the p-adic expansion of the elements in O is either finite or periodic.

Aging of the population is a rising problem for many nations worldwide. What emerges from the trend is an increasing degree of demand for eldercare. Safety of the livingalone elders at home is one of the major concerns. There are in fact commercial services available that utilize an electronic device to enable easy summon of help in an emergency. The state of the art solution infers an unusual inactivity, i.e., auto-detection of an emergency, by a long period of observation on the elder's activity level. This, although avoids false alarms, hinders timely responses to emergencies and jeopardizes the elder's health in a long run. Motivated to enable timely emergency alarm, we propose a multilevel sensor data fusion scheme that infers inactivity of an elder based on accelerometer data collected from multiple locations and time points. The preliminary results indicate that the scheme improves significantly the accuracy of inactivity inference per time point. As a result, the time period required to infer inactive state of the elder is shortened by an order of magnitude.1