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We introduce a new version v1.1 of NeatIBP. In this new version, a Kira interface is included. It allows the user to reduce the integration-by-parts (IBP) identity systems generated by NeatIBP using Kira in a highly automated way. This new version also implements the so-called spanning cuts method. It helps to reduce the total computational complexity of IBP reduction for certain hard problems. Another important feature of this new version is an algorithm to simplify the solution module of the syzygy equations hinted by the idea of maximal cuts., Comment: 57pages, 5 tables, 7 figures

In the neighborhood growth dynamics on a Hamming rectangle $[0,m-1]\times[0,n-1]\subseteq \mathbb{Z}_+^2$, the decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. After the initially occupied set is chosen, the synchronous rule is iterated. The Young domination number with a fixed latency $L$ is the smallest cardinality of an initial set that covers the rectangle by $L$ steps, for $L=0,1,\ldots$ We compute this number for some special cases, including $k$-domination for any $k$ when $m=n$, and devise approximation algorithms in the general case. These results have implications in extremal graph theory, via an equivalence between the case $L = 1$ and bipartite Tur\'an numbers for families of double stars. Our approach is based on a variety of techniques including duality, algebraic formulations, explicit constructions, and dynamic programming.

The successful deployment of deep learning-based techniques for autonomous systems is highly dependent on the data availability for the respective system in its deployment environment. Especially for unstructured outdoor environments, very few datasets exist for even fewer robotic platforms and scenarios. In an earlier work, we presented the German Outdoor and Offroad Dataset (GOOSE) framework along with 10000 multimodal frames from an offroad vehicle to enhance the perception capabilities in unstructured environments. In this work, we address the generalizability of the GOOSE framework. To accomplish this, we open-source the GOOSE-Ex dataset, which contains additional 5000 labeled multimodal frames from various completely different environments, recorded on a robotic excavator and a quadruped platform. We perform a comprehensive analysis of the semantic segmentation performance on different platforms and sensor modalities in unseen environments. In addition, we demonstrate how the combined datasets can be utilized for different downstream applications or competitions such as offroad navigation, object manipulation or scene completion. The dataset, its platform documentation and pre-trained state-of-the-art models for offroad perception will be made available on https://goose-dataset.de/. \, Comment: Submitted to IEEE for review

We consider three-state cellular automata in two dimensions in which two colored states, blue and red, compete for control of the empty background, starting from low initial densities $p$ and $q$. When the dynamics of both colored types are one-dimensional, the dynamics has three distinct phases, characterized by a power relationship between $p$ and $q$: two in which one of the colors is prevalent, and one when the colored types block each other and leave most of the space forever empty. When one of the colors spread in two dimensions and the other in one dimension, we also establish a power relation between $p$ and $q$ that characterizes which of the two colors eventually controls most of the space.

We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the computation of syzygies, then apply these techniques to deal with ideal and ring theoretic concepts such as primary decomposition and normalization, and finally use them for geometric case studies which concern curves and surfaces, both from a local and global point of view., Comment: 50 pages, 10 figures

We derive the notions of BV unital infinitesimal bialgebra and BV Frobenius algebra from the topology of suitable compactifications of moduli spaces of decorated genus 0 curves. We construct these structures respectively on reduced symplectic homology and Rabinowitz Floer homology. As an application, we construct these structures in nonequivariant string topology. We also show how the Lie bialgebra structure in equivariant string topology, and more generally on $S^1$-equivariant symplectic homology, is obtained as a formal consequence., Comment: 76 pages, including 1 appendix

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R}) $, $n\geq 3$, that preserve the minus partial order in both directions., Comment: 20 pages, 6 figures

In this survey paper we present known results about reflexive subspace lattices. We show that every nest and every atomic Boolean subspace lattice in a complex Banach space is reflexive, even strongly reflexive. Our main tool is Ringrose's Lemma about presence of rank-one operators in a reflexive algebra of operators. We consider realizations of small lattices as subspace lattices in Banach spaces. We exhibit some realizations of the pentagon and prove that some of these realizations are reflexive. On the other hand, we show that double triangle subspace lattices in finite-dimensional Banach spaces are non-reflexive., Comment: 26 pages, 5 fugures, survey paper

Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is unbounded with respect to modality and Milnor number. This method is based on proving that all right-equivalence classes within a mu-constant stratum can be represented by a single normal form derived from a regular basis of a suitably selected special fiber. While both Arnold's and our preceding work on normal forms addresses the determination of a normal form family containing the given germ, this paper takes the next natural step: We present an algorithm for computing for a given germ the values of the moduli parameters in its normal form family, that is, a normal form equation in its stable equivalence class. This algorithm will be crucial for understanding the moduli stacks of such singularities. The implementation of this algorithm, along with the foundational classification techniques, is implemented in the library arnold.lib for the computer algebra system Singular., Comment: 28 pages

We answer a question about the diameter of an order-super-commuting graph on a symmetric group by studying the number-theoretical concept of $d$-complete sequences of primes in arithmetic progression.

Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift the results to the rationals. We present a novel, massively parallel framework for modular computations with polynomial data, which is able to cover a broad spectrum of applications in commutative algebra and algebraic geometry. We demonstrate the framework's effectiveness in Groebner basis computations over the rationals and algorithmic methods from birational geometry. In particular, we develop algorithms to compute images and domains of rational maps, as well as determining invertibility and computing inverses. Our implementation is based on the Singular/GPI-Space framework, which uses the computer algebra system Singular as computational backend, while coordination and communication of parallel computations is handled by the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language. Convenient installation is realized through the package manager Spack. Relying on Petri nets, our approach provides automated parallelization and balancing of the load between computation, lifting, stabilization testing, and potential verification. We use error tolerant rational reconstruction to ensure termination as long as for a fixed computation there exist only finitely many bad primes. Via stabilization testing, our approach automatically finds with high probablity a minimal set of primes required for the successful reconstruction. We present timings to illustrate the potential for a game changing improvement of performance over previous modular and non-modular methods. In particular, we illustrate that the approach scales very well with the number of processor cores used for the computation., Comment: 56 pages, 10 figures, 6 tables

We investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents who may have different entitlements. We characterize the existence of a connected strongly-proportional allocation -- one in which every agent receives a contiguous piece worth strictly more than their proportional share. The characterization is supplemented with an algorithm that determines its existence using O(n * 2^n) queries. We devise a simpler characterization for agents with strictly positive valuations and with equal entitlements, and present an algorithm to determine the existence of such an allocation using O(n^2) queries. We provide matching lower bounds in the number of queries for both algorithms. When a connected strongly-proportional allocation exists, we show that it can also be computed using a similar number of queries. We also consider the problem of deciding the existence of a connected allocation of a cake in which each agent receives a piece worth a small fixed value more than their proportional share, and the problem of deciding the existence of a connected strongly-proportional allocation of a pie., Comment: Full version of paper accepted to ECAI 2024. Added pie allocation, stronger-than-strongly-proportional allocation, and fixed typographic issues

We present an enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. We have implemented the algorithm both using Singular and OSCAR. The implementations outperform by far the current method provided in Singular. The OSCAR implementation, benefiting in particular from just-in-time compilation, again by far outperforms the implementation of the new algorithm in Singular. This advancement in computing the Gromov-Witten invariants facilitates a study of number theoretic and geometric properties of the generating series, including quasi-modularity and homogeneity., Comment: 20 pages, 5 figures, 2 tables. arXiv admin note: text overlap with arXiv:1309.5893

The potential for deploying autonomous systems can be significantly increased by improving the perception and interpretation of the environment. However, the development of deep learning-based techniques for autonomous systems in unstructured outdoor environments poses challenges due to limited data availability for training and testing. To address this gap, we present the German Outdoor and Offroad Dataset (GOOSE), a comprehensive dataset specifically designed for unstructured outdoor environments. The GOOSE dataset incorporates 10 000 labeled pairs of images and point clouds, which are utilized to train a range of state-of-the-art segmentation models on both image and point cloud data. We open source the dataset, along with an ontology for unstructured terrain, as well as dataset standards and guidelines. This initiative aims to establish a common framework, enabling the seamless inclusion of existing datasets and a fast way to enhance the perception capabilities of various robots operating in unstructured environments. The dataset, pre-trained models for offroad perception, and additional documentation can be found at https://goose-dataset.de/., Comment: Preprint; Submitted to IEEE for review

Preclinical research and organ-dedicated applications require high-resolution positron emission tomography (PET) detectors to visualize small structures and understand biological processes at a finer level of detail. Current commercial systems often employ finely pixelated or monolithic scintillators, each with its limitations. We present a semi-monolithic detector, tailored for high-resolution PET applications, and merging concepts of monolithic and pixelated crystals. The detector features slabs measuring (24 x 10 x 1) sq. mm, coupled to a 12 x 12 readout channel photosensor with 4 mm pitch. The slabs are grouped in two arrays of 44 slabs each to achieve a higher optical photon density. We employ a fan beam collimator for fast calibration to train machine-learning-based positioning models for all three dimensions, including slab identification and depth-of-interaction (DOI), utilizing gradient tree boosting (GTB). Energy calculation was based on a position-dependent energy calibration. Using an analytical timing calibration, time skews were corrected for coincidence timing resolution (CTR) estimation. Leveraging machine-learning-based calibration in all three dimensions, we achieved high detector spatial resolution: down to 1.18 mm full width at half maximum (FWHM) detector spatial resolution and 0.75 mm mean absolute error (MAE) in the planar-monolithic direction along the slabs, and 2.14 mm FWHM and 1.03 mm MAE for depth-of-interaction (DOI) at an energy window of (435-585) keV. Correct slab interaction identification exceeded 80%, alongside an energy resolution of 13.8% and a CTR of 450 ps FWHM. Therewith, the introduced finely segmented, high-resolution slab detector demonstrates an appealing performance suitable for high-resolution PET applications. The current benchtop-based detector calibration routine allows these detectors to be used in PET systems., Comment: 14 pages, 11 figures, IEEE NSS MIC RTSD 2023

Lead-halide perovskite nanocrystals such as CsPbBr$_3$, exhibit efficient photoluminescence (PL) up-conversion, also referred to as anti-Stokes photoluminescence (ASPL). This is a phenomenon where irradiating nanocrystals up to 100 meV below gap results in higher energy band edge emission. Most surprising is that ASPL efficiencies approach unity and involve single photon interactions with multiple phonons. This is unexpected given the statistically disfavored nature of multiple-phonon absorption. Here, we report and rationalize near-unity anti-Stokes photoluminescence efficiencies in CsPbBr$_3$ nanocrystals and attribute it to resonant multiple-phonon absorption by polarons. The theory explains paradoxically large efficiencies for intrinsically disfavored, multiple-phonon-assisted ASPL in nanocrystals. Moreover, the developed microscopic mechanism has immediate and important implications for applications of ASPL towards condensed phase optical refrigeration.