arXiv subjects

Christian Weiß

Publications and source records attributed to Christian Weiß.

35 records · Page 2Linked to original sources

arXiv2021

Remarks on the Pair Correlation Statistic of Kronecker Sequences and Lattice Point Counting

In this short note, we reformulate the task of calculating the pair correlation statistics of a Kronecker sequence as a lattice point counting problem. This can be done analogously to the lattice based approach which was used to (re-)prove the famous three gap property for Kronecker sequences. We show that recently developed lattice point counting techniques can then be applied to derive that a certain class of Kronecker sequences have $β$-pair correlations for all $0 < β< 1$.

math.NT↗

arXiv2021

Discrepancy Properties and Conjugacy Classes of Interval Exchange Transformations

Interval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange transformations with low-discrepancy orbits are known so far and only for $n=2,3$ intervals, there are criteria to completely characterize those interval exchange transformations. In this paper, it is shown that having low-discrepancy orbits is a conjugacy class invariant under composition of maps. To a certain extent, this approach allows us to distinguish interval exchange transformations with low-discrepancy orbits from those without. For $n=4$ intervals, the classification is almost complete with the only exceptional case having monodromy invariant $ρ= (4,3,2,1)$. This particular monodromy invariant is discussed in detail.

math.NT↗

arXiv2021

Systems of Rank One, Explicit Rokhlin Towers, and Covering Numbers

Rotations $f_α$ of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle $α$ are known to be dynamical systems of rank one. This is equivalent to the property that the covering number $F^*(f_α)$ of the dynamical system is one. In other words, there exists a basis $B$ such that for arbitrarily high $h$ an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower $(f_α^kB)_{k=0}^{h-1}$. Although $B$ can be chosen with diameter smaller than any fixed $\varepsilon > 0$, it is not always possible to take an interval for $B$ but this can only be done when the partial quotients of $α$ are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when $B$ is the union of $n_B \in \mathbb{N}$ disjoint intervals. This question has been answered in the case $n_B =1$ by Checkhova, and here we address the general situation. If $n_B = 2$ we give a precise formula for the maximum proportion. Furthermore, we show that for fixed $α$ the maximum proportion converges to $1$ when $n_B \to \infty$. Explicit lower bounds can be given if $α$ has constant partial quotients. Our approach is inspired by the construction involved in the proof of the Rokhlin Lemma and furthermore makes use of the Three Gap Theorem.

math.DS↗

arXiv2021

Multi-dimensional Kronecker Sequences with a Small Number of Gap Lengths

Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum metric. It was proved that a generic multi-dimensional Kronecker attains the maximal possible number of different gap lengths for every sub-exponential subsequence. We mirror this result in dimension $d \in \left\{ 2, 3 \right\}$ by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely $N \in \mathbb{N}$. Our proof relies on simple arguments from the theory of continued fractions.

math.NT↗

arXiv2021

A Generalized Faulhaber Inequality, Improved Bracketing Covers, and Applications to Discrepancy

We prove a generalized Faulhaber inequality to bound the sums of the $j$-th powers of the first $n$ (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.

math.CO↗

arXiv2021

Sequences with almost Poissonian Pair Correlations

Although a generic uniformly distributed sequence has Poissonian pair correlations, only one explicit example has been found up to now. Additionally, it is even known that many classes of uniformly distributed sequences, like van der Corput sequences, Kronecker sequences and LS sequences, do not have Poissonian pair correlations. In this paper, we show that van der Corput sequences and the Kronecker sequence for the golden mean are as close to having Poissonian pair correlations as possible: they both have $α$-pair correlations for all $0 < α< 1$ but not for $α= 1$ which corresponds to Poissonian pair correlations.

math.NT↗

arXiv2020

Scheduling a Proportionate Flow Shop of Batching Machines

In this paper we study a proportionate flow shop of batching machines with release dates and a fixed number $m \geq 2$ of machines. The scheduling problem has so far barely received any attention in the literature, but recently its importance has increased significantly, due to applications in the industrial scaling of modern bio-medicine production processes. We show that for any fixed number of machines, the makespan and the sum of completion times can be minimized in polynomial time. Furthermore, we show that the obtained algorithm can also be used to minimize the weighted total completion time, maximum lateness, total tardiness and (weighted) number of late jobs in polynomial time if all release dates are $0$. Previously, polynomial time algorithms have only been known for two machines.

cs.DS↗

arXiv2020

Families of well approximable measures

We provide an algorithm to approximate a finitely supported discrete measure $μ$ by a measure $ν_{N}$ corresponding to a set of $N$ points so that the total variation between $μ$ and $ν_N$ has an upper bound. As a consequence if $μ$ is a (finite or infinitely supported) discrete probability measure on $[0,1]^{d}$ with a sufficient decay rate on the weights of each point, then $μ$ can be approximated by $ν_N$ with total variation, and hence star-discrepancy, bounded above by $(\log N) N^{-1}$. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure $μ$, there exist finite sets whose star-discrepancy with respect to $μ$ is at most $(\log N)^{d-\frac{1}{2}} N^{-1}$. Moreover we close a gap in the literature for discrepancy in the case $d=1$ showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

math.NT↗

arXiv2019

Grouping of Contracts in Insurance using Neural Networks

Despite the high importance of grouping in practice, there exists little research on the respective topic. The present work presents a complete framework for grouping and a novel method to optimize model points. Model points are used to substitute clusters of contracts in an insurance portfolio and thus yield a smaller, computationally less burdensome portfolio. This grouped portfolio is controlled to have similar characteristics as the original portfolio. We provide numerical results for term life insurance and defined contribution plans, which indicate the superiority of our approach compared to K-means clustering, a common baseline algorithm for grouping. Lastly, we show that the presented concept can optimize a fixed number of model points for the entire portfolio simultaneously. This eliminates the need for any pre-clustering of the portfolio, e.g. by K-means clustering, and therefore presents our method as an entirely new and independent methodology.

q-fin.RM↗

arXiv2019

An Aspect of Optimal Regression Design for LSMC

Practitioners sometimes suggest to use a combination of Sobol sequences and orthonormal polynomials when applying an LSMC algorithm for evaluation of option prices or in the context of risk capital calculation under the Solvency II regime. In this paper, we give a theoretical justification why good implementations of an LSMC algorithm should indeed combine these two features in order to assure numerical stability. Moreover, an explicit bound for the number of outer scenarios necessary to guarantee a prescribed degree of numerical stability is derived. We embed our observations into a coherent presentation of the theoretical background of LSMC in the insurance setting.

q-fin.CP↗

arXiv2019

Deducing Three Gap Theorem From Rauzy-Veech Induction

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places $n$ points on a circle, at angles of $z, 2z, 3z, \ldots nz$ from the starting point. The theorem was first proven in 1958 by Sós and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.

math.DS↗

arXiv2018

Improving a Constant in High-Dimensional Discrepancy Estimates

For all $s \geq 1$ and $N \geq 1$ there exist sequences $(z_1,\ldots,z_N)$ in $[0,1]^s$ such that the star-discrepancy of these points can be bounded by $$D_N^*(z_1,\ldots,z_N) \leq c \frac{\sqrt{s}}{\sqrt{N}}.$$ The best known value for the constant is $c=10$ as has been calculated by Aistleitner in \cite{Ais11}. In this paper we improve the bound to $c=9$.

math.NT↗

arXiv2018

Interval Exchange Transformations and Low-Discrepancy

In [Mas82] and [Vee78] it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case of $n=2$ intervals corresponds to circle rotation, where conditions for low-discrepancy are well-known. In this paper, we give corresponding conditions in the case $n=3$. Furthermore, we construct infinitely many interval exchange transformations with low-discrepancy orbits for $n \geq 4$. We also show that these examples do not coincide with $LS$-sequences if $S \geq 2$.

math.NT↗

arXiv2018

The Influence of Seed Selection on the Solvency II Ratio

This article contains the first published example of a real economic balance sheet where the Solvency II ratio substantially depends on the seed selected for the random number generator (RNG) used. The theoretical background and the main quality criteria for RNGs are explained in detail. To serve as a gauge for RNGs, a definition of true randomness is given. Quality tests that RNGs should pass in order to generate stable results when used in risk management under Solvency II are described.

q-fin.RM↗

arXiv2017

On the Classification of LS-Sequences

This paper adresses the question whether the $LS$-sequences constructed by Carbone yield indeed a new family of low discrepancy sequences. While it is well known that the case $S=0$ corresponds to van der Corput sequences, we prove here that the case $S=1$ can be traced back to two-sided Kronecker sequences and moreover that for $S \geq 2$ none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for $S=1$ and $L$ arbitrary.

math.NT↗

arXiv2017

Non-congruence of homology Veech groups in genus two

We study the action of the Veech group of square-tiled surfaces of genus two on homology. This action defines the homology Veech group which is a subgroup of $\textrm{SL}_2(\mathcal{O}_D)$ where $\mathcal{O}_D$ is a quadratic order of square discriminant. Extending a result of Weitze-Schmithüsen we show that also the homology Veech group is a totally non-congruence subgroup with exceptions stemming only from the prime ideals lying above 2. While Weitze-Schmithüsen's result for Veech groups is asymmetric with respect to the spin structure our use of the homology Veech group yields a completely symmetric picture.

math.GT↗

arXiv2013

Twisted Teichmüller curves

Let $X_D$ denote the Hilbert modular surface $\HH \times \HH^- / \SL_2(\OD)$. In \cite{HZ76}, F. Hirzebruch and D. Zagier introduced Hirzebruch-Zagier cycles, that could also be called twisted diagonals. These are maps $\HH \to \HH \times \HH^-$ given by $z \mapsto (Mz,-M^σz)$ where $M \in \GL_2^+(K)$ and $σ$ denotes the Galois conjugate. The projection of a twisted diagonal to $X_D$ yields a Kobayashi curve, i.e. an algebraic curve which is a geodesic for the Kobayashi metric on $X_D$. Properties of Hirzebruch-Zagier cycles have been abundantly studied in the literature.\\Teichmüller curves are algebraic curves in the moduli space of Riemann surfaces $\mathcal{M}_g$, which are geodesic for the Kobayashi metric. Some Teichmüller curves in $\mathcal{M}_2$, namely the primitive ones, can also be regarded as Kobayashi curves on $X_D$. This implies that in the universal cover the curve is of the form $z \mapsto (z,φ(z))$ for some holomorphic map $φ$. A possibility to construct even more Kobayashi curves on $X_D$ is to consider the projection of $(Mz,M^σφ(z))$ to $X_D$ where again $M \in \GL_2^+(K)$. These new objects are called twisted Teichmüller curves because their construction reminds very much of twisted diagonals. In these notes we analyze twisted Teichmüller curves in detail and describe some of their main properties. In particular, we calculate their volume and partially classify components.

math.AG↗