相关论文: Probabilistic Lower Bounds for the Discrepancy of …

Given any full rank lattice and a natural number N , we regard the point set given by the scaled lattice intersected with the unit square under the Lambert map to the unit sphere, and show that its spherical cap discrepancy is at most of…

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a…

In this paper we provide a method to obtain tight lower bounds on the minimum redundancy achievable by a Huffman code when the probability distribution underlying an alphabet is only partially known. In particular, we address the case where…

We propose upper and lower bounds on the maximum success probability for discriminating given quantum states. The proposed upper bound is obtained from a suboptimal solution to the dual problem of the corresponding optimal state…

We derive bounds for the ball $L_p$-discrepancies in the Hamming space for $0<p<\infty$ and $p=\infty$. Sharp estimates of discrepancies have been obtained for many spaces such as the Euclidean spheres and more general compact Riemannian…

We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbf{\Omega}=(\Omega_1,\ldots,\Omega_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be…

The L_2-discrepancy measures the irregularity of the distribution of a finite point set. In this note we prove lower bounds for the L_2 discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper…

For 100 years since galaxies were found to be flying apart from each other, astronomers have been trying to determine how fast. The expansion, characterized by the Hubble constant, H0, is confused locally by peculiar velocities caused by…

1. We find that any two distance moduli measurements for the same galaxy differ from each other by 2.07 times the reported one sigma uncertainty on average. 2. This average difference between distance moduli measurements of the same galaxy…

There is an approximately 9% discrepancy, corresponding to 2.4sigma, between two independent constraints on the expansion rate of the universe: one indirectly arising from the cosmic microwave background and baryon acoustic oscillations,…

Stratified sampling is a fast and simple method to generate point sets with uniform distribution in hypercubes. However, for the most common paraxial stratfication it has the prominent drawback that the number of sampled points in n…

Points in the unit cube with low discrepancy can be constructed using algebra or, more recently, by direct computational optimization of a criterion. The usual $L_\infty$ star discrepancy is a poor criterion for this because it is…

Computer simulations serve as powerful tools for scientists and engineers to gain insights into complex systems. Less costly than physical experiments, computer experiments sometimes involve large number of trials. Conventional design…

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely…

The construction and implementation of atmospheric model grids is a popular tool in exoplanet characterisation. These typically vary a number of parameters linearly, containing one model for every combination of parameter values. Here we…

Building upon the exact methods presented in our earlier work [J. Complexity, 2022], we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing…

There are irreducible differences between the Hubble constant measured locally and the global value. They are due to density perturbations and finite sample volume (cosmic variance) and finite number of objects in the sample (sampling…

Magnitude predictions of $\Lambda$CDM, as parametrized by the Planck collaboration, are not consistent with the supernova data of the whole Pantheon+ sample even when, in order to take into account the uncertainty about its value, the…

We establish the existence of $N$-point sets in dimension $d$ whose star-discrepancy is bounded above by $2.4631832 \sqrt{\frac{d}{N}}$, where the numerical constant improves upon all previously known bounds. This improvement is obtained by…

Sample complexity of bias estimation is a lower bound on the runtime of any bias detection method. Many regulatory frameworks require the bias to be tested for all subgroups, whose number grows exponentially with the number of protected…