arXiv subjects

Christian Haase

Publications and source records attributed to Christian Haase.

At least 19 recordsLinked to original sources

arXiv2026

Containments of Tensor Network Varieties

Building upon the work of Buczy\'nska et al., we study here tensor formats and their corresponding encoding of tensors via two-fold tensor products determined by the combinatorics of a binary tree. The set of all tensors representable by a given network forms the corresponding tensor network variety. A very basic question asks whether every tensor representable by one network is representable by another network, namely, when one tensor network variety is contained in another. Specific instances of this question became known as the Hackbusch Conjecture. Here, we propose a general framework for this question and take first steps, theoretical as well as experimental, towards a better understanding. In particular, given any two binary trees on $n$ leaves, we define (and prove existence of) a new measure, the containment exponent, which gauges how much one has to boost the parameters of one network for the containment to hold. We present an algorithm for bounding these containment exponents of tensor network varieties and report on an exhaustive search among trees on up to $n=8$ leaves.

math.CO↗

arXiv2026

Classifying the Fine Polyhedral Spectrum

In this paper, we examine an analogue of the recently solved spectrum conjecture by Fujita in the setting of Fine polyhedral adjunction theory. We present computational results for lower-dimensional polytopes, which lead to a complete classification of the highest numbers of this Fine spectrum in any dimension. Moreover, we present a classification of the Fine spectrum in dimensions one, two and (almost) three, while providing a framework for general classification results in any dimension.

math.CO↗

arXiv2025

Multigraded Betti numbers of Veronese embeddings

In this paper, we study the multigraded Betti numbers of Veronese embeddings of projective spaces. Due to Hochster's formula, we interpret these multigraded Betti numbers in terms of the homology of certain simplicial complexes. By analyzing these simplicial complexes and applying Forman's discrete Morse theory, we derive vanishing and non-vanishing results for these multigraded Betti numbers.

math.AC↗

arXiv2023

LDP polygons and the number 12 revisited

We give a combinatorial proof of a lattice point identity involving a lattice polygon and its dual, generalizing the formula $area(\Delta) + area(\Delta^*) = 6$ for reflexive $\Delta$. The identity is equivalent to the stringy Libgober-Wood identity for toric log del Pezzo surfaces.

math.CO↗

arXiv2023

Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes

We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized volume of faces of such polytopes.

cs.LG↗

arXiv2023

Fine Polyhedral Adjunction Theory

Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces, the Fine interior of a lattice polytope got recently into the focus of research. It is has been used for constructing canonical models in the sense of Mori Theory [arXiv:2008.05814]. Based on the Fine interior, we propose here a modification of the original adjoint polytopes as defined in [arXiv:1105.2415], by defining the Fine adjoint polytope $P^{F(s)}$ of $P$ as consisting of the points in $P$ that have lattice distance at least $s$ to all valid inequalities for $P$. We obtain a Fine Polyhedral Adjunction Theory that is, in many respects, better behaved than its original analogue. Many existing results in Polyhedral Adjunction Theory carry over, some with stronger conclusions, as decomposing polytopes into Cayley sums, and most with simpler, more natural proofs as in the case of the finiteness of the Fine spectrum.

math.CO↗

arXiv2022

On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

math.AG↗

arXiv2022

METL: a modern ETL pipeline with a dynamic mapping matrix

Modern ETL streaming pipelines extract data from various sources and forward it to multiple consumers, such as data warehouses (DW) and analytical systems that leverage machine learning (ML). However, the increasing number of systems that are connected to such pipelines requires new solutions for data integration. The canonical (or common) data model (CDM) offers such an integration. It is particular useful for integrating microservice systems into ETL pipelines. (Villaca et al 2020, Oliveira et al 2019) However, a mapping to a CDM is complex. (Lemcke et al 2012) There are three complexity problems, namely the size of the required mapping matrix, the automation of updates of the matrix in response to changes in the extraction sources and the time efficiency of the mapping. In this paper, we present a new solution for these problems. More precisely, we present a new dynamic mapping matrix (DMM), which is based on permutation matrices that are obtained by block-partitioning the full mapping matrix. We show that the DMM can be used for automated updates in response to schema changes, for parallel computation in near real-time and for highly efficient compacting. For the solution, we draw on research into matrix partitioning (Quinn 2004) and dynamic networks (Haase et al 2021). The DMM has been implemented into an app called Message ETL (METL). METL is the key part of a new ETL streaming pipeline at EOS that conducts the transformation to a CDM. The ETL pipeline is based on Kafka-streams. It extracts data from more than 80 microservices with log-based Change Data Capture (CDC) with Debezium and loads the data to a DW and an ML platform. EOS is part of the Otto-Group, the second-largest e-commerce provider in Europe.

cs.DC↗

arXiv2022

SCoT: Sense Clustering over Time: a tool for the analysis of lexical change

We present Sense Clustering over Time (SCoT), a novel network-based tool for analysing lexical change. SCoT represents the meanings of a word as clusters of similar words. It visualises their formation, change, and demise. There are two main approaches to the exploration of dynamic networks: the discrete one compares a series of clustered graphs from separate points in time. The continuous one analyses the changes of one dynamic network over a time-span. SCoT offers a new hybrid solution. First, it aggregates time-stamped documents into intervals and calculates one sense graph per discrete interval. Then, it merges the static graphs to a new type of dynamic semantic neighbourhood graph over time. The resulting sense clusters offer uniquely detailed insights into lexical change over continuous intervals with model transparency and provenance. SCoT has been successfully used in a European study on the changing meaning of `crisis'.

cs.CL↗

arXiv2021

Competitive equilibrium always exists for combinatorial auctions with graphical pricing schemes

We show that a competitive equilibrium always exists in combinatorial auctions with anonymous graphical valuations and pricing, using discrete geometry. This is an intuitive and easy-to-construct class of valuations that can model both complementarity and substitutes, and to our knowledge, it is the first class besides gross substitutes that have guaranteed competitive equilibrium. We prove through counter-examples that our result is tight, and we give explicit algorithms for constructive competitive pricing vectors. We also give extensions to multi-unit combinatorial auctions (also known as product-mix auctions). Combined with theorems on graphical valuations and pricing equilibrium of Candogan, Ozdagar and Parrilo, our results indicate that quadratic pricing is a highly practical method to run combinatorial auctions.

math.CO↗

arXiv2021

Control of Electrochemical Corrosion Properties by Influencing Mn Partitioning through Intercritically Annealing of Medium-Mn Steel

Medium-Mn steels exhibit excellent mechanical properties and lower production costs compared to high-Mn steels, which makes them a potential material for future application in the automotive industry. Intercritical annealing (ICA) after cold rolling allows to control the stacking fault energy (SFE) of austenite, the fraction of ferrite and reverted austenite, and the element partitioning (especially Mn). Although Mn deteriorates the corrosion behavior of Fe-Mn-Al alloys, the influence of austenite fraction and element partitioning of Mn on the electrochemical corrosion behavior has not been investigated yet. Therefore, the electrochemical corrosion behavior in 0.1 M H2SO4 of X6MnAl12-3, which was intercritically annealed for 2 h at 550 °C, 600 °C and 700 °C, was investigated by potentiodynamic polarization (PDP), electrochemical impedance spectroscopy (EIS) and mass spectroscopy with inductively coupled plasma (ICP-MS). Additionally, specimens after 1 h and 24 h of immersion were examined via SEM to visualize the corrosion damage. The ICA specimens showed a selective dissolution of reverted austenite due to its micro-galvanic coupling with the adjacent ferrite. The severity of the micro-galvanic coupling can be reduced by decreasing the interface area as well as the chemical gradient of mainly Mn between ferrite and reverted austenite by ICA.

cond-mat.mtrl-sci↗

arXiv2021

Classifier construction in Boolean networks using algebraic methods

We investigate how classifiers for Boolean networks (BNs) can be constructed and modified under constraints. A typical constraint is to observe only states in attractors or even more specifically steady states of BNs. Steady states of BNs are one of the most interesting features for application. Large models can possess many steady states. In the typical scenario motivating this paper we start from a Boolean model with a given classification of the state space into phenotypes defined by high-level readout components. In order to link molecular biomarkers with experimental design, we search for alternative components suitable for the given classification task. This is useful for modelers of regulatory networks for suggesting experiments and measurements based on their models. It can also help to explain causal relations between components and phenotypes. To tackle this problem we need to use the structure of the BN and the constraints. This calls for an algebraic approach. Indeed we demonstrate that this problem can be reformulated into the language of algebraic geometry. While already interesting in itself, this allows us to use Groebner bases to construct an algorithm for finding such classifiers. We demonstrate the usefulness of this algorithm as a proof of concept on a model with 25 components.

math.AC↗

arXiv2021

Toric Newton-Okounkov functions with an application to the rationality of certain Seshadri constants on surfaces

We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before.

math.AG↗

arXiv2021

The Finiteness Threshold Width of Lattice Polytopes

We prove that in each dimension $d$ there is a constant $w^\infty(d)\in \mathbb{N}$ such that for every $n\in \mathbb{N}$ all but finitely many $d$-polytopes with $n$ lattice points have width at most $w^\infty(d)$. We call $w^\infty(d)$ the finiteness threshold width and show that $d-2 \le w^\infty(d)\le O^*\left( d^{4/3}\right)$. Blanco and Santos determined the value $w^\infty(3)=1$. Here, we establish $w^\infty(4)=2$. This implies, in particular, that there are only finitely many empty $4$-simplices of width larger than two. The main tool in our proofs is the study of $d$-dimensional lifts of hollow $(d-1)$-polytopes.

math.CO↗

arXiv2019

Algebraic Hyperbolicity for Surfaces in Toric Threefolds

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.

math.AG↗

arXiv2019

3D aspects of materials design for metals and alloys

Nowadays, most structural integrity concepts rely on simplified isotropic ma-terial data that are used within continuum mechanics modeling approaches. In contrast, modern casting and forming processes yield complex microstruc-tures coming along with pronounced gradients in the material's properties in various length scales. By means of additive manufacturing. bionics-inspired structures can be perfectly adapted to the space-resolved load distribution. Specific temperature control during solidification or heat treatment allow for usage of anisotropy and precipitation effects for materials optimization down to the atomic length scale. The present paper shows the significance and practical application of a three-dimensional correlation between the material's microstructure and its technical properties.

cond-mat.mtrl-sci↗

arXiv2019

Application-oriented strain-hardening engineering of high-manganese steels

The outstanding mechanical properties of high-manganese steels (HMnS) are a result of their high strain-hardenability. That is facilitated by strong suppression of dynamic recovery, predominant planar glide, and the activation of additional deformation mechanisms, such as transformation-induced plasticity (TRIP) and twinning-induced plasticity (TWIP). However, depending on the final application, strongly differing requirements on the mechanical properties of HMnS are of relevance. In order to design HMnS for specific applications, multi-scale material simulation under consideration of the processing conditions that allows for prediction of the final mechanical properties is required, i.e. strain-hardening engineering based on integrated computational materials engineering (ICME). In this work, we present an approach that employs alloy selection by stacking-fault energy calculations, which enables activation/suppression of specific deformation mechanisms. Tailored manufacturing, e.g. by thermo-mechanical treatment, severe plastic deformation or additive manufacturing, provides possibilities to make use of the strain-hardenability during and after processing to define the mechanical properties. A computational approach for alloy and process design will be discussed.

cond-mat.mtrl-sci↗

arXiv2019

Maximum Number of Modes of Gaussian Mixtures

Gaussian mixture models are widely used in Statistics. A fundamental aspect of these distributions is the study of the local maxima of the density, or modes. In particular, it is not known how many modes a mixture of $k$ Gaussians in $d$ dimensions can have. We give a brief account of this problem's history. Then, we give improved lower bounds and the first upper bound on the maximum number of modes, provided it is finite.

math.ST↗