Searcharxiv⌕ Search

arXiv subjects

Stephan Baier

Publications and source records attributed to Stephan Baier.

At least 55 records · Page 3Linked to original sources
arXiv2021

Diophantine Approximation with Prime Restriction in Real Quadratic Number Fields

The distribution of $αp$ modulo one, where $p$ runs over the rational primes and $α$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $ν>0$ one can establish the infinitude of primes $p$ satisfying $||αp||\le p^{-ν}$. The latest record in this regard is Kaisa Matomäki's landmark result $ν=1/3-\varepsilon$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog for $\mathbb{Q}(i)$ of his result in the context of $\mathbb{Q}$, which yields an exponent of $ν=7/22$. Marc Technau and the first-named author produced an analogue of Bob Vaughan's result $ν=1/4-\varepsilon$ for all imaginary quadratic number fields of class number 1. In the present article, we establish an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction. This setting involves the additional complication of an infinite group of units in the ring of integers. Moreover, although the basic sieve approach remains the same (we use an ideal version of Harman's sieve), the problem takes a different flavor since it becomes truly 2-dimensional. We reduce it eventually to a counting problem which is, interestingly, related to roots of quadratic congruences. To approximate them, we use an approach by Christopher Hooley based on the theory of binary quadratic forms.

math.NT↗
arXiv2020

Conformance checking: A state-of-the-art literature review

Conformance checking is a set of process mining functions that compare process instances with a given process model. It identifies deviations between the process instances' actual behaviour ("as-is") and its modelled behaviour ("to-be"). Especially in the context of analyzing compliance in organizations, it is currently gaining momentum -- e.g. for auditors. Researchers have proposed a variety of conformance checking techniques that are geared towards certain process model notations or specific applications such as process model evaluation. This article reviews a set of conformance checking techniques described in 37 scholarly publications. It classifies the techniques along the dimensions "modelling language", "algorithm type", "quality metric", and "perspective" using a concept matrix so that the techniques can be better accessed by practitioners and researchers. The matrix highlights the dimensions where extant research concentrates and where blind spots exist. For instance, process miners use declarative process modelling languages often, but applications in conformance checking are rare. Likewise, process mining can investigate process roles or process metrics such as duration, but conformance checking techniques narrow on analyzing control-flow. Future research may construct techniques that support these neglected approaches to conformance checking.

cs.SE↗
arXiv2020

Large sieve with sparse sets of moduli for $\mathbb{Z}[i]$

We establish a general large sieve inequality with sparse sets $\mathcal{S}$ of moduli in the Gaussian integers which are in a sense well-distributed in arithmetic progressions. This extends earlier work of S. Baier on the large sieve with sparse sets of moduli. We then use this result to obtain large sieve bounds for the cases when $\mathcal{S}$ consists of squares of Gaussian integers and of Gaussian primes. Our bound for the case of square moduli improves a recent result by the authors.

math.NT↗
arXiv2019

Prime powers dividing products of consecutive integer values of $x^{2^n}+1$

Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every positive integer $m$, there exists a prime divisor $p$ of $P_{m,2}$ such that ord$_{p} (P_{m,2}) \leq 4$. Consequently, $P_{m,2}$ is never a fifth or higher power. This extends work of Cilleruelo who studied the case $n=1$.

math.NT↗
arXiv2019

Large sieve inequality with power moduli for function fields

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high dimensions and for power moduli.

math.NT↗
arXiv2019

Divisibility problems for function fields

We investigate three combinatorial problems considered by Erdös, Rivat, Sarközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.

math.NT↗
arXiv2019

Central limit theorems for elliptic curves and modular forms with smooth weight functions

The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a Central Limit Theorem for much smaller families of modular forms, and we succeed in proving a theorem of this type for families of elliptic curves under the Riemann Hypothesis for $L$-functions associated to Hecke eigenforms for the full modular group.

math.NT↗
arXiv2019

Analyzing Customer Feedback for Product Fit Prediction

One of the biggest hurdles for customers when purchasing fashion online, is the difficulty of finding products with the right fit. In order to provide a better online shopping experience, platforms need to find ways to recommend the right product sizes and the best fitting products to their customers. These recommendation systems, however, require customer feedback in order to estimate the most suitable sizing options. Such feedback is rare and often only available as natural text. In this paper, we examine the extraction of product fit feedback from customer reviews using natural language processing techniques. In particular, we compare traditional methods with more recent transfer learning techniques for text classification, and analyze their results. Our evaluation shows, that the transfer learning approach ULMFit is not only comparatively fast to train, but also achieves highest accuracy on this task. The integration of the extracted information with actual size recommendation systems is left for future work.

cs.CL↗
arXiv2018

Improving Visual Relationship Detection using Semantic Modeling of Scene Descriptions

Structured scene descriptions of images are useful for the automatic processing and querying of large image databases. We show how the combination of a semantic and a visual statistical model can improve on the task of mapping images to their associated scene description. In this paper we consider scene descriptions which are represented as a set of triples (subject, predicate, object), where each triple consists of a pair of visual objects, which appear in the image, and the relationship between them (e.g. man-riding-elephant, man-wearing-hat). We combine a standard visual model for object detection, based on convolutional neural networks, with a latent variable model for link prediction. We apply multiple state-of-the-art link prediction methods and compare their capability for visual relationship detection. One of the main advantages of link prediction methods is that they can also generalize to triples, which have never been observed in the training data. Our experimental results on the recently published Stanford Visual Relationship dataset, a challenging real world dataset, show that the integration of a semantic model using link prediction methods can significantly improve the results for visual relationship detection. Our combined approach achieves superior performance compared to the state-of-the-art method from the Stanford computer vision group.

cs.CL↗
arXiv2018

Improving Information Extraction from Images with Learned Semantic Models

Many applications require an understanding of an image that goes beyond the simple detection and classification of its objects. In particular, a great deal of semantic information is carried in the relationships between objects. We have previously shown that the combination of a visual model and a statistical semantic prior model can improve on the task of mapping images to their associated scene description. In this paper, we review the model and compare it to a novel conditional multi-way model for visual relationship detection, which does not include an explicitly trained visual prior model. We also discuss potential relationships between the proposed methods and memory models of the human brain.

cs.AI↗
arXiv2018

Applications of the square sieve to a conjecture of Lang and Trotter for a pair of elliptic curves over the rationals

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $p \neq \ell$, the Frobenius automorphism associated to $p$ (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of $\ell$. Its splitting field is called the Frobenius field of $E$ at $p$. Let $E_1$ and $E_2$ be two elliptic curves defined over $\mathbb{Q}$ that are non-isogenous over $\overline{\mathbb{Q}}$ and also without complex multiplication over $\overline{\mathbb{Q}}$. In analogy with the well-known Lang-Trotter conjecture for a single elliptic curve, it is natural to consider the asymptotic behaviour of the function that counts the number of primes $p \leq x$ such that the Frobenius fields of $E_1$ and $E_2$ at $p$ coincide. In this short note, using Heath-Brown's square sieve, we provide both conditional (upon the Generalized Riemann Hypothesis) and unconditional upper bounds.

math.NT↗
arXiv2018

Moments of the error term in the Sato-Tate law for elliptic curves

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. Our estimates are stronger than those obtained by W.D. Banks and I.E. Shparlinski (arXiv:math/0609144) and L. Zhao and the fist-named author in (arXiv:math/0608318) for the first and second moments, but this comes at the cost of larger ranges of averaging. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work by the second-named author and K. Sinha (arXiv:1705.04115) who derived a Central Limit Theorem on the distribution of the errors in the Sato-Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch's identities connect moments of coefficients of Hasse-Weil $L$-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature.

math.NT↗
arXiv2018

The large sieve with power moduli for $\mathbb{Z}[i]$

We establish a large sieve inequality for power moduli in $\mathbb{Z}[i]$, extending earlier work by L. Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in $\mathbb{Z}$. Our method starts with a version of the large sieve for $\mathbb{R}^2$. We convert the resulting counting problem back into one for $\mathbb{Z}[i]$ which we then attack using Weyl differencing and Poisson summation.

math.NT↗
arXiv2017

Attention-based Information Fusion using Multi-Encoder-Decoder Recurrent Neural Networks

With the rising number of interconnected devices and sensors, modeling distributed sensor networks is of increasing interest. Recurrent neural networks (RNN) are considered particularly well suited for modeling sensory and streaming data. When predicting future behavior, incorporating information from neighboring sensor stations is often beneficial. We propose a new RNN based architecture for context specific information fusion across multiple spatially distributed sensor stations. Hereby, latent representations of multiple local models, each modeling one sensor station, are jointed and weighted, according to their importance for the prediction. The particular importance is assessed depending on the current context using a separate attention function. We demonstrate the effectiveness of our model on three different real-world sensor network datasets.

cs.LG↗
arXiv2017

Tensor Decompositions for Modeling Inverse Dynamics

Modeling inverse dynamics is crucial for accurate feedforward robot control. The model computes the necessary joint torques, to perform a desired movement. The highly non-linear inverse function of the dynamical system can be approximated using regression techniques. We propose as regression method a tensor decomposition model that exploits the inherent three-way interaction of positions x velocities x accelerations. Most work in tensor factorization has addressed the decomposition of dense tensors. In this paper, we build upon the decomposition of sparse tensors, with only small amounts of nonzero entries. The decomposition of sparse tensors has successfully been used in relational learning, e.g., the modeling of large knowledge graphs. Recently, the approach has been extended to multi-class classification with discrete input variables. Representing the data in high dimensional sparse tensors enables the approximation of complex highly non-linear functions. In this paper we show how the decomposition of sparse tensors can be applied to regression problems. Furthermore, we extend the method to continuous inputs, by learning a mapping from the continuous inputs to the latent representations of the tensor decomposition, using basis functions. We evaluate our proposed model on a dataset with trajectories from a seven degrees of freedom SARCOS robot arm. Our experimental results show superior performance of the proposed functional tensor model, compared to challenging state-of-the art methods.

cs.LG↗