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The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in R n . The VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the Fundamental Theorem of Calculus, which we call the Cone Differentiation Theorem.

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula, discuss its stability properties and demonstrate its efficiency on numerical examples. As an unexpected bonus of our approach, we prove a long-standing conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

Given a weighted graph G(V, E) and t ≥ 1, a subgraph H is a t-spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t. The subsetwise spanner problem aims to preserve distances in G for only a subset of the vertices. We generalize the minimum-cost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the multi-level graph spanner (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multi-level graph visualization, especially where vertices may require different grades of service. We formulate a 0-1 integer linear program (ILP) of size O(|E||V | 2 ) for the more general minimum pairwise spanner problem, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomial-size ILP. We extend this ILP formulation to the MLGS problem, and evaluate the heuristic and ILP performance on random graphs of up to 100 vertices and 500 edges.

The aim of this study has been to predict the track geometry degradation rate using artificial neural network. Tack geometry measurements, asset information, and maintenance history for five line sections from the Swedish railway network were collected, processed, and prepared to develop the ANN model. The information of track was taken into account and different features of track sections were considered as model input variables. In addition, Garson method was applied to explore the relative importance of the variables affecting geometry degradation rate. By analysing the performance of the model, we found out that the ANN has an acceptable capability in explaining the variability of degradation rates in different locations of the track. In addition, it is found that the maintenance history, the degradation level after tamping, and the frequency of trains passing along the track have the strongest contributions among the considered set of features in prediction of degradation rate.