We show how standard Newtonian N-body simulations can be interpreted in terms of the weak-field limit of general relativity by employing the recently developed Newtonian motion gauge. Our framework allows the inclusion of radiation perturbations and the non-linear evolution of matter. We show how to construct the weak-field metric by combining Newtonian simulations with results from Einstein-Boltzmann codes. We discuss observational effects on weak lensing and ray tracing, identifying important relativistic corrections.
In this section we discuss the connection between our work, the analytic method of @cite_33 , the relativistic ray tracing code liger and the relativistic N-body code gevolution .
@cite_33
2015816463 1910014366 2103374230 2760946714
We present algorithms to reconstruct the planar cross-section of a simply connected object from data points measured by rays. The rays are semi-infinite curves representing, for example, the laser beam or the articulated arms of a robot moving around the object. This paper shows that the information provided by the rays is crucial (though generally neglected) when solving 2-dimensional reconstruction problems. The main property of the rays is that they induce a total order on the measured points. This order is shown to be computable in optimal time O(n log n). The algorithm is fully dynamic and allows the insertion or the deletion of a point in O(log n) time. From this order a polygonal approximation of the object can be deduced in a straightforward manner. However, if insufficient data are available or if the points belong to several connected objects, this polygonal approximation may not be a simple polygon or may intersect the rays. This can be checked in O(n log n) time. The order induced by the rays can also be used to find a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, we have shown that 3n-3 probes are necessary and sufficient if the object has n non-colinear edges. Each probe can be determined in O(log n) time yielding an O(n log n)-time 0(n)-space algorithm. When each probe outcome consists of a contact point and a ray measuring that point but not the normal, the same strategy can still be applied. Under a mild condition, 8n-4 probes are sufficient to discover a shape that is almost surely the actual shape of the object. We present a new approach to detection and tracking of moving objects with a 2D laser scanner for autonomous driving applications. Objects are modelled with a set of rigidly attached sample points along their boundaries whose positions are initialized with and updated by raw laser measurements, thus allowing a non-parametric representation that is capable of representing objects independent of their classes and shapes. Detection and tracking of such object models are handled in a theoretically principled manner as a Bayes filter where the motion states and shape information of all objects are represented as a part of a joint state which includes in addition the pose of the sensor and geometry of the static part of the world. We derive the prediction and observation models for the evolution of the joint state, and describe how the knowledge of the static local background helps in identifying dynamic objects from static ones in a principled and straightforward way. Dealing with raw laser points poses a significant challenge to data association. We propose a hierarchical approach, and present a new variant of the well-known Joint Compatibility Branch and Bound algorithm to respect and take advantage of the constraints of the problem introduced through correlations between observations. Finally, we calibrate the system systematically on real world data containing 7,500 labelled object examples and validate on 6,000 test cases. We demonstrate its performance over an existing industry standard targeted at the same problem domain as well as a classical approach to model-free object tracking. We introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel E sub b N sub 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece (1996). Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. Previous studies have demonstrated the utility and applicability of machine learning techniques to jet physics. In this paper, we construct new observables for the discrimination of jets from different originating particles exclusively from information identified by the machine. The approach we propose is to first organize information in the jet by resolved phase space and determine the effective @math -body phase space at which discrimination power saturates. This then allows for the construction of a discrimination observable from the @math -body phase space coordinates. A general form of this observable can be expressed with numerous parameters that are chosen so that the observable maximizes the signal vs. background likelihood. Here, we illustrate this technique applied to discrimination of @math decays from massive @math splittings. We show that for a simple parametrization, we can construct an observable that has discrimination power comparable to, or better than, widely-used observables motivated from theory considerations. For the case of jets on which modified mass-drop tagger grooming is applied, the observable that the machine learns is essentially the angle of the dominant gluon emission off of the @math pair.