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We study small non-trivial solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, with $q$ being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli $q$. Above, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume that $(\alpha_2\alpha_3,q)=1$. We show that for all $\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$ is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients., Comment: 14 Pages

We prove that for every irrational number $\alpha$, real number $\beta$, real number $c$ satisfying $1

In this paper, we establish sharp upper bounds on shifted moments of the family of Dirichlet $L$-functions to a fixed modulus over function fields. We apply the result to obtain upper bounds on moments of Dirichlet character sums over function fields., Comment: 9 pages

We present an elementary uniform lower bound for the classical Kloosterman sum $S(a,b;c)$ under the condition of its non-vanishing and $(ab,c)=1$, with $c$ being an odd integer. We then apply this lower bound for Kloosterman sums to derive an explicit lower bound in the Petersson's trace formula, subject to a pertinent condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, wherein the parameters $k$ and $N$ are permitted to vary independently., Comment: 11 pages

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $\lambda_1x_1^2+\cdots +\lambda_nx_n^2\equiv \lambda_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $\lambda_1,...,\lambda_{n+1}$ are integer coefficients such that $(\lambda_1\cdots \lambda _{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $\lambda_1,...,\lambda_n>0$ and $(\lambda_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\varepsilon)m}$, centered at the origin. If $n\ge 4$ and $\lambda_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $\lambda_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag., Comment: 20 pages

We consider small solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume that $(\alpha_2\alpha_3,q)=1$. We show that for all $\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\varepsilon}$ as $q$ tends to infinity over the set of all odd prime powers. It is of significance that we break the barrier 1/2 in the above exponent. If $q$ is restricted to powers $p^m$ of a {\it fixed} prime $p$ and $m$ tends to infinity, we obtain a slight improvement of this result using the theory of $p$-adic exponent pairs, as developed by Mili\'cevi\'c, replacing the exponent $11/24$ above by $11/25$. Under the Lindel\"of hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$., Comment: 10 pages

Matom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke $L$-functions., Comment: 31 pages

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of quadratic congruences of the form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus establishing an equidistribution result for these solutions., Comment: Major changes as compared to our first version, 10 pages

In this note, we give a summary of the article ``The distribution of prime ideals of imaginary quadratic fields'' by G. Harman, A. Kumchev and P. A. Lewis and establish analogous results for real quadratic fields based on the same method., Comment: 12 pages

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k$th-powers. Secondly, we input a recent and general result of Wooley on mean values of exponential sums. Lastly, we state a conditional result, based on the conjectural Hardy-Littlewood formula for the number of representations of a large positive integer as a sum of $k+1$ $k$th-powers., Comment: 13 pages, added theorem 2.4 part ii, added corollary 2.6, new comparison section

In the thirties of the last century, I. M. Vinogradov proved that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matom\"aki (2009) who showed the infinitude of prime solutions of the inequality $||p\alpha||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. Recently, in \cite{BaMo}, the authors established an analogue of Matom\"aki's result for imaginary quadratic extensions of the function field $k=\mathbb{F}_q(T)$. In this paper, we consider the case of real quadratic extensions of $k$ of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent $\theta=1/4$). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper \cite{BaMo} on the imaginary quadratic case. We also simplify arguments in the paper \cite{BM} on the same problem for real quadratic number fields by D. Mazumder and the first-named author., Comment: 27 pages

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le (\log x)^C$ to sparse sets of moduli $s\le x^{1/3-\varepsilon}$ with radical rad$(s)\le x^{9/40}$. To derive our result, we combine our previous method with a Bombieri-Vinogradov-type theorem for general moduli $s\le x^{9/40}$ obtained by Roger Baker., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:2107.04348

Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for solutions with $x_3\asymp M$, where $M$ is much smaller than $\sqrt{D}$. To be precise, their condition is $M\ge D^{1/2-1/1332}$. Their analysis led them to averages of certain Weyl sums. The condition of $D$ being square-free is essential in their work. We investigate the "opposite" case when $D=n^2$ is a square of an odd integer $n$. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with $x_3\asymp M$, where $M\ge D^{1/2-1/16+\varepsilon}$ for any fixed $\varepsilon>0$., Comment: 30 pages

Keating and Rudnick derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus $Q$ is a polynomial in $\mathbb{F}_q[T]$ such that $Q(0)\not=0$. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition., Comment: 10 pages

We translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields., Comment: It has been brought to our attention that our result is a special case of Theorem 2.1 in the following paper: Chih-Nung Zhu, "Diophantine inequalities for the non-Archimedean line F_q((1/T))", Acta Arith. 97, No. 3, 253-267 (2001). We were not aware of this work. Further important work in this direction was done by Craig Spencer

Let $p>5$ be a fixed prime and assume that $\alpha_1,\alpha_2,\alpha_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $\alpha_1,\alpha_2,\alpha_3$ are fixed and $n\rightarrow \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are allowed to vary with $n$, we show that this formula holds if $N\gg q^{11/18+\varepsilon}$. The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $N \gg q^{5/8+\varepsilon}$ for odd square-free moduli $q$., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:2201.05871

Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{\nu n}$ with $\nu>1/2$., Comment: 8 pages

We implement an iterative numerical method to solve polynomial equations $f(x)=0$ in the $p$-adic numbers, where $f(x) \in\mathbb{Z}_p[x]$. This method is a simplified $p$-adic analogue of Jarratt's method for finding roots of functions over the real numbers. We establish that our method has a higher order of convergence than J.F.T. Rabago's $p$-adic version of Olver's method from 2016. Moreover, we weaken the initial conditions in Rabago's method, which allows us to start the iteration with a multiple root of the congruence $f(x) \equiv 0 \bmod{p}$., Comment: 9 pages

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $p\alpha$ when $\alpha$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matom\"aki (2009) who showed the infinitude of prime solutions of the inequality $||p\alpha||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. We prove function field analogues of this result for the fields $k=\mathbb{F}_q(T)$ and imaginary quadratic extensions $K$ of $k$. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly., Comment: 28 pages

In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose cardinality is bounded by a power of $\log x$. We use a $p$-adic variant Harman's sieve to extend Baker's range to $q\le x^{1/4-\varepsilon}$ if $\mathcal{S}$ is restricted to prime powers $p^N$, where $p\le (\log x)^C$ for some fixed but arbitrary $C>0$. For large enough $C$, we thus get an almost all result. Previously, an asymptotic estimate for $\pi(x;p^N,a)$ of the expected kind, with $p$ being an odd prime, was established in the wider range $p^N\le x^{3/8-\varepsilon}$ by Barban, Linnik and Chudakov \cite{BLC}. Gallagher \cite{Gal} extended this range to $p^N\le x^{2/5-\varepsilon}$ and Huxley \cite{Hux2} improved Gallagher's exponent to $5/12$. A lower bound of the correct order of magnitude was recently established by Banks and Shparlinski \cite{BaS} for the even wider range $p^N\le x^{0.4736}$. However, all these results hold for {\it fixed} primes $p$, and the $O$-constants in the relevant estimates depend on $p$. Therefore, they do not contain our result. In a part of our article, we describe how our method relates to these results., Comment: 22 pages

We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent $1/4$ obtained by the first and third named authors for imaginary quadratic number fields \cite{BT} and by the first and second named authors for real quadratic number fields \cite{BM} to $7/22$. This generalizes a result of Harman \cite{HarZi} who obtained the same exponent $7/22$ for $\mathbb{Q}(i)$ by extending his method which gave this exponent for $\mathbb{Q}$ \cite{harman1996on-the-distribu}. Our proof is based on an extension of his sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in \cite{BM}, for which we use analytic properties of Hecke L-functions with Gr\"o\ss encharacters., Comment: 39 pages

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.

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-\nu}$. The latest record in this regard is Kaisa Matom\"aki's landmark result $\nu=1/3-\varepsilon$ 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 $\nu=7/22$. Marc Technau and the first-named author produced an analogue of Bob Vaughan's result $\nu=1/4-\varepsilon$ 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., Comment: 38 pages

In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics., Comment: 14 pages

We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound obtained in arXiv:1802.03131 by Baier and Singh. Indeed, we point out an error in arXiv:1802.03131., Comment: 42 pages

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.

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., Comment: 24 pages

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 $n=1$., Comment: 12 pages

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula., Comment: 36 pages

We prove a lower bound for the large sieve with square moduli., Comment: To appear in Bull. Aust. Math. Soc

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., Comment: 18 pages

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.

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.

We investigate three combinatorial problems considered by Erd\"os, Rivat, Sark\"ozy and Sch\"on 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., Comment: 12 pages. We added an erratum to the original article

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., Comment: 17 pages

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., Comment: 14 pages. The authors have detected an error in this article. A description of this error and a corrected version of the large sieve for square moduli in function fields is given in their article arXiv:1910.05043

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.

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.

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., Comment: 15 pages

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $p\gamma$, $p$ running over the primes and $\gamma$ being a fixed irrational, for Gaussian primes., Comment: 32 pages. This is a combination of slightly extended versions of my papers arXiv:1609.08745 and arXiv:1612.02622, with the goal to make arXiv:1612.02622 completely self-contained

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., Comment: 26 pages

We establish a Polya-Vinogradov-type bound for finite periodic multipicative characters on the Gaussian integers., Comment: This result has been proved in stronger and much more general form by E. Landau using Hecke L-functions, which escaped my attention. However, the main part of the method in my article consists of refined estimates for averages of linear exponential sums on Z[i] over regions. These estimates have other applications, and I will provide one soon in a new paper

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes., Comment: 16 pages