Explicit lower bounds for the length of the shortest opaque set for the unit disc and the unit square in the Euclidean plane are derived. The results are based on an explicit application of the general method of Kawamura, Moriyama, Otachi and Pach [9]. Employing a recent observation by Steinerberger on the possible orientations of straight barriers with length close to Jones’ bound, we improve the bound in [9] by more than a factor 3. The bound for barriers of the unit disc is new and based on the idea that the free parameters in the general method from [9] can be optimized due to the strong symmetry properties of the disc. Our approach illustrates both the power and the limitations of the method.

We study the symmetry groups and winding numbers of planar curves obtained as images of weighted sums of exponentials. More generally, we study the image of the complex unit circle under a finite or infinite Laurent series using a particular parametrization of the circle. We generalize various previous results on such sums of exponentials and relate them to other classes of curves present in the literature. Moreover, we consider the evolution under the wave equation of such curves for the case of binomials. Interestingly, our methods provide a unified and systematic way of constructing curves with prescribed properties, such as the number of cusps, the number of intersection points or the winding number. Comment: 24 pages, 7 figures, Journal version

Classical jittered sampling partitions $[0,1]^d$ into $m^d$ cubes for a positive integer $m$ and randomly places a point inside each of them, providing a point set of size $N=m^d$ with small discrepancy. The aim of this note is to provide a construction of partitions that works for arbitrary $N$ and improves straight-forward constructions. We show how to construct equivolume partitions of the $d$-dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube. We investigate the discrepancy of such point sets and optimise the expected discrepancy numerically by relaxing the equivolume constraint using different black-box optimisation techniques.

This paper addresses the challenge of extending general finite sequences of real numbers within a subinterval of the real line, maintaining their inherent statistical properties by employing machine learning. Our focus lies on preserving the gap distribution and pair correlation function of these point sets. Leveraging advancements in deep learning applied to point processes, this paper explores the use of an auto-regressive \textit{Sequence Extension Mixture Model} (SEMM) for extending finite sequences, by estimating directly the conditional density, instead of the intensity function. We perform comparative experiments on multiple types of point processes, including Poisson, locally attractive, and locally repelling sequences, and we perform a case study on the prediction of Riemann $\zeta$ function zeroes. The results indicate that the proposed mixture model outperforms traditional neural network architectures in sequence extension with the retention of statistical properties. Given this motivation, we showcase the capabilities of a mixture model to extend sequences, maintaining specific statistical properties, i.e. the gap distribution, and pair correlation indicators.

We study the expected $\mathcal{L}_2$-discrepancy of stratified samples generated from special equi-volume partitions of the unit square. The partitions are defined via parallel lines that are all orthogonal to the diagonal of the square. It is shown that the expected discrepancy of stratified samples derived from these partitions is a factor 2 smaller than the expected discrepancy of the same number of i.i.d uniformly distributed random points in the unit square. We conjecture that this is best possible among all partitions generated from parallel lines.

The automotive industry is facing the challenge of gaining knowledge from large datasets—originating for example from the research and development process, IT systems, production, or from fleet data. Problems most likely become manifest in data and result in increased costs. Examples can range from problems in the on-board electrical system to virtual validation of autonomous driving functions in R&D. Hence, one important use case is the automatic detection of anomalous behavior in the data to forecast and identify potential problems as early as possible. We demonstrate new mathematical methods from Topological Data Analysis (TDA) that can help to address these kinds of problems. TDA is a rather new field in mathematics that combines techniques from geometry and topology to analyze noisy datasets. Beside academia, it has been applied successfully to various fields including medicine (identification of tumor cells), finance (fraud detection), and materials science (structure analysis). We highlight two main methods from TDA: the (ball) mapper algorithm and persistent homology and illustrate potential applications in automotive industry. We illustrate these abstract methods and show that they can produce valuable knowledge about potential problems—for example in the automotive context—giving an added value to the customer and the OEM.

Abstract For m, d ∈ ℕ, a jittered sample of N = m d points can be constructed by partitioning [0, 1]d into m d axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any m, d ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.

We prove that classical jittered sampling of the d -dimensional unit cube does not yield the smallest expected L 2-discrepancy among all stratified samples with N = m d points. Our counterexample can be given explicitly and consists of convex partitioning sets of equal volume.

In this note we are interested in the rich geometry of the graph of a curve $\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*} \gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which $a,b$ are two different positive integers. It turns out that the sum of only two exponentials gives already rise to intriguing graphs. We determine the symmetry group and the points of self intersection of any such graph using only elementary arguments and describe various interesting phenomena that arise in the study of graphs of sums of more than two exponentials.

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 chosen uniformly in the $i$th set of the partition (and stochastically independent of the other points), $i=1,\ldots,N$. For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected ${\mathcal{L}_p}$-discrepancy, $\mathbb{E} {\mathcal{L}_p}(\mathcal{P}_{\mathbf{\Omega}})^p$, of a point set $\mathcal{P}_\mathbf{\Omega}$ generated from any equivolume partition $\mathbf{\Omega}$ is always strictly smaller than the expected ${\mathcal{L}_p}$-discrepancy of a set of $N$ uniform random samples for $p>1$. For fixed $N$ we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected ${\mathcal{L}_p}$-discrepancy. We illustrate our results with explicit constructions for small $N$. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every $N$.

• True random number generators are often preferred in high security applications. • AIS31 evaluation criteria require stochastic model to test unpredictability of TRNG. • We study self-timed ring based TRNG and the related stochastic model. • We improve entropy bounds for such TRNGs. • Our results can lower hardware requirements and stimulate further research. We improve entropy bounds for a self-timed ring based true random number generator, taking the timing of the reference clock signals into account. The models we discuss encompass both perfect and jittered reference clocks. Importantly, our novel analysis of jittered reference clocks can be used to study how robust the improved entropy bounds are. We use parity filters as post-processing blocks and improve results on the required minimal parity filter size to obtain a given target entropy. In addition, we see in numerical experiments that these models are robust in the sense that the minimal required size of a parity filter to exceed a given entropy bound does not change when weakening the assumption on the reference clock; i.e. when considering jittered instead of perfect reference clocks.

We prove that classical jittered sampling of the d-dimensional unit cube does not yield the smallest expected L2-discrepancy among all stratified samples with N = m d points. Our counterexample can be given explicitly and consists of convex partitioning sets of equal volume.

For coprime integers $N,a,b,c$, with $0

This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing 'well-distributed' sequences of points on $[0,1)$. Let $f:[0,1] \rightarrow \mathbb{R}$ be (i) symmetric $f(x) = f(1-x)$, (ii) twice differentiable on $(0,1)$, and (iii) such that $f''(x)>0$ for all $x \in (0,1)$. We study the greedy dynamical system, where, given an initial set $\{x_0, \ldots, x_{N-1}\} \subset [0,1)$, the point $x_N$ is obtained as $$ x_{N} = \arg\min_{x \in [0,1)} \sum_{k=0}^{N-1}{f(|x-x_k|)}.$$ We prove that if we start this construction with the single element $x_0=0$, then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): \textit{greedy energy minimization recovers the way we count in binary.} This gives a new construction of the classical van der Corput sequence. The special case $f(x) = 1-\log(2 \sin(\pi x))$ answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk. Moreover, we give a general bound on the discrepancy of any sequence constructed in this way for functions $f$ satisfying an additional assumption.

SummaryWe discuss an amusing application of number theory: suppose you find yourself on the two-dimensional torus T2 equipped with N candles and want to position the candles in such a way that they heat up the room as efficiently as possible. To achieve this goal, we give a construction that uses number theory as the main ingredient and explain related results.

We construct and implement a uniformly distributed sequence in the orthogonal group O(n). From this sequence we obtain a uniformly distributed sequence on the Grassmannian manifold G(n,k), which we use to approximate integral-geometric formulas. We show that our algorithm compares well with classical random constructions which motivates various directions for future research.

The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. On the occasion of Henri's 80th birthday we aim to survey (some of) his contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versus using a sequence of permutations. Throughout, we collect, sharpen and extend open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.

The nucleator is a method to estimate the volume of a particle, i.e., a compact subset of R-3, which is widely used in Stereology. It is based on geometric sampling and known to be unbiased. However, the prediction of the variance of this estimator is non-trivial and depends on the underlying sampling scheme. We propose well established tools from quasi-Monte Carlo integration to address this problem. In particular, we show how the theory of reproducing kernel Hilbert spaces can be used for variance prediction and how the variance of estimators based on the nucleator idea can be reduced using lattice (or lattice-like) points. We illustrate and test our results on various examples.

We study the persistent homology of random Čech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the kth persistent Betti number of a random Čech complex and determine its asymptotic order in the subcritical regime. This extends a result of Kahle on the asymptotic order of the ordinary kth Betti number of such complexes to the persistent setting.