Properties of parallelotopes equivalent to Voronoi's conjecture

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope.Comment: 18 pages (submitted

Problems in GPS Accuracy

Improving and predicting the accuracy of positioning estimates derived from the global positioning system (GPS) continues to be a problem of great interest. Dependable and accurate positioning is especially important for navigation applications such as the landing of commercial aircraft. This subject gives rise to many interesting and challenging mathematical problems. This dissertation investigates two such problems. The first problem involves the study of the relationship between positioning accuracy and satellite geometry configurations relative to a user\u27s position. In this work, accuracy is measured by so-called dilution of precision (DOP) terms. The DOP terms arise from the linear regression model used to estimate user position from GPS observables, and are directly related to user position errors. An analysis of the statistical properties explaining the behavior of the DOP terms is presented. The most accurate satellite geometries and worst configurations are given for some cases. The second problem involves finding methods for detecting and repairing cycle-slips in range delay data between a satellite and a receiver. The distance between a satellite and a receiver can be estimated by measuring the difference in the carrier frequency phase shift experienced between the satellite and receiver oscillators. Cycle-slips are discontinuities in the integer number of complete cycles in these data, and are caused by interruptions or degradations in the signal such as low signal to noise ratio, software failures, or physical obstruction of the signals. These slips propagate to errors in user positioning. Cycle-slip detection and repair are crucial to maintaining accurate positioning. Linear regression models and sequential hypothesis testing are used to model, detect, and repair cycle-slips. The effectiveness of these methods is studied using data obtained from ground-station receivers