The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, N ( ε , s ) , defined as the minimum number of points required to achieve a star discrepancy of at most ε in dimension s . It is known that N ( ε , s ) depends only linearly on the dimension s . Finding explicit point set constructions that achieve this optimal linear dependence on the dimension remains a major open problem. In this paper, we make progress on this question by analyzing point sets constructed from a multiset union of digitally shifted Korobov polynomial lattice point sets. Specifically, we show the following two results. A union of randomly generated Korobov polynomial lattice point sets shifted by a random digital shift of depth m can achieve a star discrepancy whose inverse depends only linearly on s . The second result shows that a union of all Korobov polynomial lattice point sets, each shifted by a different random digital shift, achieves the same star discrepancy bound. While our proof relies on a concentration result (Bernstein's inequality) and is therefore non-constructive, it significantly reduces the search space for such point sets from a continuum of possibilities to a finite set of candidates, marking a step towards a fully explicit construction.

In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the d -dimensional unit cube. We show that intractability holds for multivariate integration in the standard Wiener algebra in the deterministic setting, in contrast to polynomial tractability in an unweighted subspace of the Wiener algebra recently shown by Goda (2023). Moreover, we prove that multivariate integration in the subspace of the Wiener algebra introduced by Goda is strongly polynomially tractable if we switch to the randomized setting, where we obtain a better ε -exponent than the one implied by the standard Monte Carlo method . We also identify subspaces in which multivariate integration in the deterministic setting are (strongly) polynomially tractable and we compare these results with the bound which can be obtained via Hoeffding's inequality.

The transparent cornea is the window of the eye, facilitating the entry of light rays and controlling focusing the movement of the light within the eye. The cornea is critical, contributing to 75% of the refractive power of the eye. Keratoconus is a progressive and multifactorial corneal degenerative disease affecting 1 in 2000 individuals worldwide. Currently, there is no cure for keratoconus other than corneal transplantation for advanced stage keratoconus or corneal cross-linking, which can only halt KC progression. The ability to accurately identify subtle KC or KC progression is of vital clinical significance. To date, there has been little consensus on a useful model to classify KC patients, which therefore inhibits the ability to predict disease progression accurately.In this paper, we utilised machine learning to analyse data from 124 KC patients, including topographical and clinical variables. Both supervised multilayer perceptron and unsupervised variational autoencoder models were used to classify KC patients with reference to the existing Amsler-Krumeich (A-K) classification system. Both methods result in high accuracy, with the unsupervised method showing better performance. The result showed that the unsupervised method with a selection of 29 variables could be a powerful tool to provide an automatic classification tool for clinicians. These outcomes provide a platform for additional analysis for the progression and treatment of keratoconus.

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams-type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$-dimensional unit cube $[0,1)^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further.