Previous models for emission tomography (ET) do not distinguish the physics of ET from that of transmission tomography. We give a more accurate general mathematical model for ET where an unknown emission density lambda = lambda(x, y, z) generates, and is to be reconstructed from, the number of counts n(*)(d) in each of D detector units d. Within the model, we give an algorithm for determining an estimate lambdainsertion mark of lambda which maximizes the probability p(n(*)|lambda) of observing the actual detector count data n(*) over all possible densities lambda. Let independent Poisson variables n(b) with unknown means lambda(b), b = 1, ..., B represent the number of unobserved emissions in each of B boxes (pixels) partitioning an object containing an emitter. Suppose each emission in box b is detected in detector unit d with probability p(b, d), d = 1, ..., D with p(b,d) a one-step transition matrix, assumed known. We observe the total number n(*) = n(*)(d) of emissions in each detector unit d and want to estimate the unknown lambda = lambda(b), b = 1, ..., B. For each lambda, the observed data n(*) has probability or likelihood p(n(*)|lambda). The EM algorithm of mathematical statistics starts with an initial estimate lambda(0) and gives the following simple iterative procedure for obtaining a new estimate lambdainsertion mark(new), from an old estimate lambdainsertion mark(old), to obtain lambdainsertion mark(k), k = 1, 2, ..., lambdainsertion mark(new)(b)= lambdainsertion mark(old)(b)Sum of (n(*)p(b,d) from d=1 to D/Sum of lambdainsertion mark()old(b('))p(b('),d) from b(')=1 to B), b=1,...B.

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The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are specified? One need consider only paths with at most 2 cusps or reversals. We give a set of paths which is sufficient in the sense that it always contains a shortest path and small in the sense that there are at most 68, but usually many fewer paths in the set for any pair of endpoints and directions. We give these paths by explicit formula. Calculating the length of each of these paths and selecting the (not necessarily unique) path with smallest length yields a simple algorithm for a shortest path in each case. These optimal paths or geodesies may be described as follows: If C is an arc of a circle of the minimal turning radius and S is a line segment, then it is sufficient to consider only certain paths of the form CCSCC where arcs and segments fit smoothly, one or more of the arcs or segments may vanish, and where reversals, or equivalently cusps, between arcs or segments are allowed. This contrasts with the case when cusps are not allowed, where Dubins (1957) has shown that paths of the form CCC and CSC suffice. Introduction.We want to find a shortest path in the plane with specified initial and final points and directions and with the further constraint that at each point the radius of curvature should be > 1. This problem arose in a simple model for a robot cart which moves under computer control. The cart can shift into reverse and so the path is allowed to have cusps.In an elegant paper, Lester Dubins (1957) solved the problem when the car cannot reverse and cusps are not allowed. Even in this case it is apparently impossible to give an explicit formula for the shortest path. Instead Dubins gives a sufficient set of paths, i.e. a set which always contains what he called a geodesic, or optimal path. His sufficient set is so small that there are at most 6 contenders in the set for each case of specified endpoint conditions, and it is a simple matter to find the shortest of these 6, which gives an algorithm for the solution. He showed that any geodesic can be described by one of 6 words:

In this paper, we introduce a mathematical framework for the characterization of network interference in wireless systems. We consider a network in which the interferers are scattered according to a spatial Poisson process and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We start by determining the statistical distribution of the aggregate network interference. We then investigate four applications of the proposed model: 1) interference in cognitive radio networks; 2) interference in wireless packet networks; 3) spectrum of the aggregate radio-frequency emission of wireless networks; and 4) coexistence between ultrawideband and narrowband systems. Our framework accounts for all the essential physical parameters that affect network interference, such as the wireless propagation effects, the transmission technology, the spatial density of interferers, and the transmitted power of the interferers.

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