arXiv subjects
Publications and source records attributed to Stephan Baier.
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 $pγ$, $p$ running over the primes and $γ$ being a fixed irrational, for Gaussian primes.
We establish a Polya-Vinogradov-type bound for finite periodic multipicative characters on the Gaussian integers.
It is proved that Epstein's zeta-function $ζ_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + iγ$ with $ T \leq γ\leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.
We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
We give a short alternative proof using Heath-Brown's square sieve of a bound of the author for the large sieve with square moduli.
We prove a large sieve inequality for square norm moduli in Z[i].
Embedding learning, a.k.a. representation learning, has been shown to be able to model large-scale semantic knowledge graphs. A key concept is a mapping of the knowledge graph to a tensor representation whose entries are predicted by models using latent representations of generalized entities. Latent variable models are well suited to deal with the high dimensionality and sparsity of typical knowledge graphs. In recent publications the embedding models were extended to also consider time evolutions, time patterns and subsymbolic representations. In this paper we map embedding models, which were developed purely as solutions to technical problems for modelling temporal knowledge graphs, to various cognitive memory functions, in particular to semantic and concept memory, episodic memory, sensory memory, short-term memory, and working memory. We discuss learning, query answering, the path from sensory input to semantic decoding, and the relationship between episodic memory and semantic memory. We introduce a number of hypotheses on human memory that can be derived from the developed mathematical models.
Embedding learning, a.k.a. representation learning, has been shown to be able to model large-scale semantic knowledge graphs. A key concept is a mapping of the knowledge graph to a tensor representation whose entries are predicted by models using latent representations of generalized entities. Knowledge graphs are typically treated as static: A knowledge graph grows more links when more facts become available but the ground truth values associated with links is considered time invariant. In this paper we address the issue of knowledge graphs where triple states depend on time. We assume that changes in the knowledge graph always arrive in form of events, in the sense that the events are the gateway to the knowledge graph. We train an event prediction model which uses both knowledge graph background information and information on recent events. By predicting future events, we also predict likely changes in the knowledge graph and thus obtain a model for the evolution of the knowledge graph as well. Our experiments demonstrate that our approach performs well in a clinical application, a recommendation engine and a sensor network application.
Large knowledge graphs increasingly add value to various applications that require machines to recognize and understand queries and their semantics, as in search or question answering systems. Latent variable models have increasingly gained attention for the statistical modeling of knowledge graphs, showing promising results in tasks related to knowledge graph completion and cleaning. Besides storing facts about the world, schema-based knowledge graphs are backed by rich semantic descriptions of entities and relation-types that allow machines to understand the notion of things and their semantic relationships. In this work, we study how type-constraints can generally support the statistical modeling with latent variable models. More precisely, we integrated prior knowledge in form of type-constraints in various state of the art latent variable approaches. Our experimental results show that prior knowledge on relation-types significantly improves these models up to 77% in link-prediction tasks. The achieved improvements are especially prominent when a low model complexity is enforced, a crucial requirement when these models are applied to very large datasets. Unfortunately, type-constraints are neither always available nor always complete e.g., they can become fuzzy when entities lack proper typing. We show that in these cases, it can be beneficial to apply a local closed-world assumption that approximates the semantics of relation-types based on observations made in the data.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $Δ=D/16$, where D is the discriminant, by T.D. Browning and the author. To achieve this improvement, we elaborate on our methods for counting weighted solutions of inhomogeneous cubic congruences to power-ful moduli. The novelty lies in going a step further in the explicit evaluation of complete exponential sums and saving a factor by averaging over the moduli.
We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.
Let F be an algebraically closed field of positive characteristic p. The third author and Will Turner gave an explicit description of the extension algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit basis. We examine this basis and use it to give upper and lower bounds for the growth behaviour of the dimensions of Ext-groups. Moreover, we implement an algorithm to determine the dimension of any given Ext-group.
Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.
We give an alternative proof of a recent result by T.D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.
We exhibit continua on two different growth scales in the dynamical Mertens' theorem for ergodic automorphisms of one-dimensional solenoids.
Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.