A modern aircraft may require on the order of thousands of custom shims to fill gaps between structural components in the airframe that arise due to manufacturing tolerances adding up across large structures. These shims, whether liquid or solid, are necessary to eliminate gaps, maintain structural performance, and minimize pull-down forces required to bring the aircraft into engineering nominal configuration for peak aerodynamic efficiency. Currently, gap filling is a time-consuming process, involving either expensive by-hand inspection or computations on vast quantities of measurement data from increasingly sophisticated metrology equipment. In either case, this amounts to significant delays in production, with much of the time being spent in the critical path of the aircraft assembly.In this work, we present an alternative strategy for predictive shimming, based on machine learning and sparse sensing to first learn gap distributions from historical data, and then design optimized sparse sensing strategies to streamline the collection and processing of data. This new approach is based on the assumption that patterns exist in shim distributions across aircraft, and that these patterns may be mined and used to reduce the burden of data collection and processing in future aircraft. Specifically, robust principal component analysis is used to extract low-dimensional patterns in the gap measurements while rejecting outliers. Next, optimized sparse sensors are obtained that are most informative about the dimensions of a new aircraft in these low-dimensional principal components. We demonstrate the success of the proposed approach, called PIXel Identification Despite Uncertainty in Sensor Technology (PIXI-DUST), on historical production data from 54 representative Boeing commercial aircraft. Our algorithm successfully predicts 99% of the shim gaps within the desired measurement tolerance using around 3% of the laser scan points that are typically required; all results are rigorously cross-validated.

Data science, and machine learning in particular, is rapidly transforming the scientific and industrial landscapes. The aerospace industry is poised to capitalize on big data and machine learning, which excels at solving the types of multi-objective, constrained optimization problems that arise in aircraft design and manufacturing. Indeed,

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Due to their so-called time-frequency localization properties, wavelets have become a powerful tool in signal analysis and image processing. Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. In this paper, we derive necessary and sufficient conditions for the stability of the shifts of certain compactly supported refinable functions. These conditions are in terms of the zeros of the refinement mask. Our results are actually applicable to more general distributions which are not of function type, if we generalize the notion of stability appropriately. We also provide a similar characterization of the (global) linear independence of the shifts. We present several examples illustrating our results, as well as one example in which known results on box splines are derived using the theorems of this paper. Academic Press

Abstract. Re nement equations involving matrix masks are receiving a lot of attention these days.They can play a central role in the study of re nable nitely generated shift-invariant spaces, multiresolutions generated by more than one function, multi-wavelets, splines with multiple knots, and matrix subdivision schemes | including Hermite-type subdivision schemes. Several recent papers on this subject begin with an assumption on the eigenstructure of the mask, pointing out that this assumption is heuristically natural" or preferred". In this note, we prove that stability of the shifts of the re nable function requires this assumption.

ABSTRACT. Typical constructions of wavelets depend on the stability of the shifts of an underlying re nable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g. symmetry and piecewise polynomial structure. Presently, m ultiwavelets seem to o er a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several simultaneously re nable functions. In Section 2 of this paper, we c haracterize stability and linear independence of the shifts of a nite re nable function set in terms of the re nement mask. Several illustrative examples are provided. The characterizations given in Section 2 actually require that the re nable functions be minimal in some sense. This notion of minimality is made clear in Section 3, where we provide su cient conditions on the mask to ensure minimality. The conditions are shown to be also necessary under further assumptions on the re nement mask. An example is provided illustrating how the software package MAPLE can be used to investigate at least the case of two simultaneously re nable functions.

Abstract. Refinable functions are an intrinsic part of subdivision schemes and wavelet constructions. The relevant properties of such functions must usually be determined from their refinement masks. In this paper, we provide a characterization of linear independence for the shifts of a multivariate refinable vector of distributions in terms of its (finitely supported) refinement mask.

Tverberg's theorem says that a set with sufficiently many points in R d can always be partitioned into m parts so that the (m − 1)-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg's theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve complexes, as soon as the number of points is large enough. We prove that, given a set with sufficiently many points, all trees and all cycles can also be induced by at least one partition of the point set. We also discuss how some simplicial complexes can never be achieved this way, even for arbitrarily large sets of points.