Orthogonal arrays are a type of combinatorial design that emerged in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing arrays of minimum size. Kuperberg, Lovett and Peled (2017) gave a non-constructive existence proof of orthogonal arrays whose size is near-optimal (i.e., within a polynomial of Rao’s lower bound), leaving open the question of an algorithmic construction. We give the first explicit, deterministic, algorithmic construction of orthogonal arrays achieving near-optimal size for all parameters. Our construction uses algebraic geometry codes. In pseudorandomness, the notions of t-independent generators or t-independent hash functions are equivalent to orthogonal arrays. Classical constructions of t-independent hash functions are known when the size of the codomain is a prime power, but very few constructions are known for an arbitrary codomain. Our construction yields algorithmically efficient t-independent hash functions for arbitrary domain and codomain.

Osteoporosis is an age-associated disease characterised by low bone mineral density (BMD) and micro-architectural deterioration leading to enhanced fracture risk. Conventional dual-energy X-ray absorptiometry (DXA) analysis has facilitated our understanding of BMD reduction in specific regions of interest (ROIs) within the femur, but cannot resolve spatial BMD patterns nor reflect age-related changes in bone microarchitecture due to its inherent averaging of pixel BMD values into large ROIs. To address these limitations and develop a comprehensive model of involutional bone loss, this paper presents a fully automatic pipeline to build a spatio-temporal atlas of ageing bone in the proximal femur. A new technique, termed DXA region free analysis (DXA RFA), is proposed to eliminate morphological variation between DXA scans by warping each image into a reference template. To construct the atlas, we use unprocessed DXA data from Caucasian women aged 20-97 years participating in three cohort studies in Western Europe ( ,000). A novel calibration procedure, termed quantile matching regression, is proposed to integrate data from different DXA manufacturers. Pixel-wise BMD evolution with ageing was modelled using smooth quantile curves. This technique enables characterisation of spatially-complex BMD change patterns with ageing, visualised using heat-maps. Furthermore, quantile curves plotted at different pixel coordinates showed consistently different rates of bone loss at different regions within the femoral neck. Given the close relationship between spatio-temporal bone loss and osteoporotic fracture, improved understanding of the bone ageing process could lead to enhanced prognostic, preventive and therapeutic strategies for the disease.

As the size and operator complexity of a fuzzy logic system increases, computational tractability becomes a problem. There is a significant amount of parallelism in both the creation of the fuzzy rule base and in fuzzy inference. Traditional processors (CPUs) cannot take full advantage of this natural parallelism graphics processing units (GPUs) speed up rule construction and inference by utilizing up to 128 processing units operating in parallel. Normally, these processors are used to perform high speed graphics calculations for video games, movies, and other areas of intense graphical work. In this paper, a method is discussed for speeding up fuzzy logic by structuring it into a format such that it resembles the standard rendering procedure for a graphics pipeline based on rasterization.