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For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by one edge flip. The diameter ${\rm diam}(F(P))$ of this graph is subject of intensive study. For points in general position, it is between $3n/2-5$ and $2n-4$, with no improvement for 25 years. For points in convex position, it lies between $3n/2 - 5$ and $\approx1.95n$, where the lower bound was conjectured to be tight up to an additive constant and the upper bound is a recent breakthrough improvement over several bounds of the form $2n-o(n)$. In this work, we provide new upper and lower bounds on ${\rm diam}(F(P))$, mainly focusing on points in convex position. We show $14n/9 - O(1) \le {\rm diam}(F(P)) \le 5n/3 - 3$, by this disproving the conjectured upper bound of $3n/2$ for convex position, and relevantly improving both the long-standing lower bound for general position and the recent new upper bound for convex position. We complement these by showing that if one of $T,T'$ has at most two boundary edges, then ${\rm dist}(T,T') \le 2d/2 < 3n/2$, where $d = |T-T'|$ is the number of edges in one tree that are not in the other. To prove both the upper and the lower bound, we introduce a new powerful tool. Specifically, we convert the flip distance problem for given $T,T'$ to the problem of a largest acyclic subset in an associated conflict graph $H(T,T')$. In fact, this method is powerful enough to give an equivalent formulation of the diameter of $F(P)$ for points $P$ in convex position up to lower-order terms. As such, conflict graphs are likely the key to a complete resolution of this and possibly also other reconfiguration problems., Comment: an extended abstract appears at SODA 2025

Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty$). It is well-known that computing the interleaving distance is NP-hard in both cases. We extend this result by showing that computing the $p$-presentation distance is NP-hard for all $p\in [1,\infty)$ for both merge trees and $t$-parameter persistence modules for any $t\geq 2$. Though the details differ, both proofs follow the same novel strategy, suggesting that our approach can be adapted to proving the NP-hardness of other distances based on sums or $p$-norms., Comment: 28 pages, 7 figures

We report results obtained and insights gained while answering the following question: how effective is it to use a simulator to establish path following control policies for an autonomous ground robot? While the quality of the simulator conditions the answer to this question, we found that for the simulation platform used herein, producing four control policies for path planning was straightforward once a digital twin of the controlled robot was available. The control policies established in simulation and subsequently demonstrated in the real world are PID control, MPC, and two neural network (NN) based controllers. Training the two NN controllers via imitation learning was accomplished expeditiously using seven simple maneuvers: follow three circles clockwise, follow the same circles counter-clockwise, and drive straight. A test randomization process that employs random micro-simulations is used to rank the ``goodness'' of the four control policies. The policy ranking noted in simulation correlates well with the ranking observed when the control policies were tested in the real world. The simulation platform used is publicly available and BSD3-released as open source; a public Docker image is available for reproducibility studies. It contains a dynamics engine, a sensor simulator, a ROS2 bridge, and a ROS2 autonomy stack the latter employed both in the simulator and the real world experiments., Comment: 8 pages, 7 figures

We obtain the analytical solutions to the Schr\"odinger equation for the attractive inverse-square potential in an induced electric dipole moment system under the influence of the harmonic oscillator. We show that bound states can exist when the electric field configuration brings a cut-off point that imposes a forbidden region for the neutral particle. Then, by dealing with $s$-waves, we obtain the energy eigenvalues in the strong electric field regime and for small values of the angular frequency of the harmonic oscillator. Further, we extend our discussion about the energy eigenvalues beyond the $s$-waves., Comment: 10 pages

We examine the spatial distribution of electric charges within an extended, non-conductive cylinder featuring an inner radius denoted as $r_{0}$. Our investigation unveils the emergence of a distinct modified attractive-inverse square potential, arising from the intricate interplay between the electric field and the induced electric dipole moment of a neutral particle. This modified potential notably departs from the conventional inverse-square potential, showcasing an additional term proportional to $r^{-1}$. As a result, we present compelling evidence for the realization of a discrete energy spectrum within this intricate system., Comment: 9 pages

This paper introduces DEM-Engine, a new submodule of Project Chrono, that is designed to carry out Discrete Element Method (DEM) simulations. Based on spherical primitive shapes, DEM-Engine can simulate polydisperse granular materials and handle complex shapes generated as assemblies of primitives, referred to as clumps. DEM-Engine has a multi-tier parallelized structure that is optimized to operate simultaneously on two GPUs. The code uses custom-defined data types to reduce memory footprint and increase bandwidth. A novel "delayed contact detection" algorithm allows the decoupling of the contact detection and force computation, thus splitting the workload into two asynchronous GPU streams. DEM-Engine uses just-in-time compilation to support user-defined contact force models. This paper discusses its C++ and Python interfaces and presents a variety of numerical tests, in which impact forces, complex-shaped particle flows, and a custom force model are validated considering well-known benchmark cases. Additionally, the full potential of the simulator is demonstrated for the investigation of extraterrestrial rover mobility on granular terrain. The chosen case study demonstrates that large-scale co-simulations (comprising 11 million elements) spanning 15 seconds, in conjunction with an external multi-body dynamics system, can be efficiently executed within a day. Lastly, a performance test suggests that DEM-Engine displays linear scaling up to 150 million elements on two NVIDIA A100 GPUs., Comment: 38 pages, 30 figures, 9 tables. This preprint is submitted to Computer Physics Communications

We report on a study that employs an in-house developed simulation infrastructure to accomplish zero shot policy transferability for a control policy associated with a scale autonomous vehicle. We focus on implementing policies that require no real world data to be trained (Zero-Shot Transfer), and are developed in-house as opposed to being validated by previous works. We do this by implementing a Neural Network (NN) controller that is trained only on a family of circular reference trajectories. The sensors used are RTK-GPS and IMU, the latter for providing heading. The NN controller is trained using either a human driver (via human in the loop simulation), or a Model Predictive Control (MPC) strategy. We demonstrate these two approaches in conjunction with two operation scenarios: the vehicle follows a waypoint-defined trajectory at constant speed; and the vehicle follows a speed profile that changes along the vehicle's waypoint-defined trajectory. The primary contribution of this work is the demonstration of Zero-Shot Transfer in conjunction with a novel feed-forward NN controller trained using a general purpose, in-house developed simulation platform.

High resolution spectral observations of the lower solar atmosphere (chromosphere and transition region) during coronal heating events, in combination with predictions from models of impulsively heated loops, provide powerful diagnostics of the properties of the heating in active region cores. Here we analyze the first coordinated observations of such events with the Interface Region Imaging Spectrograph (IRIS) and the CHROMospheric Imaging Spectrometer (CHROMIS), at the Swedish 1-m Solar Telescope (SST), which provided extremely high spatial resolution and revealed chromospheric brightenings with spatial dimensions down to ~150km. We use machine learning methods (k-means clustering) and find significant coherence in the spatial and temporal properties of the chromospheric spectra, suggesting, in turn, coherence in the spatial and temporal distribution of the coronal heating. The comparison of IRIS and CHROMIS spectra with simulations suggest that both non-thermal electrons with low energy (low-energy cutoff ~5keV) and direct heating in the corona transported by thermal conduction contribute to the heating of the low atmosphere. This is consistent with growing evidence that non-thermal electrons are not uncommon in small heating events (nano- to micro-flares), and that their properties can be constrained by chromospheric and transition region spectral observations., Comment: Accepted for publication in the Astrophysical Journal. 21 pages, 13 figures

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data., Comment: 26 pages, 4 figures, 9 tables; v2: final version in NeurIPS 2023

Nanoflare heating through small-scale magnetic reconnection events is one of the prime candidates to explain heating of the solar corona. However, direct signatures of nanoflares are difficult to determine, and unambiguous observational evidence is still lacking. Numerical models that include accelerated electrons and can reproduce flaring conditions are essential in understanding how low-energetic events act as a heating mechanism of the corona, and how such events are able to produce signatures in the spectral lines that can be detected through observations. We investigate the effects of accelerated electrons in synthetic spectra from a 3D radiative magnetohydrodynamics simulation to better understand small-scale heating events and their impact on the solar atmosphere. We synthesised the chromospheric Ca II and Mg II lines and the transition region Si IV resonance lines from a quiet Sun numerical simulation that includes accelerated electrons. We calculated the contribution function to the intensity to better understand how the lines are formed, and what factors are contributing to the detailed shape of the spectral profiles. The synthetic spectra are highly affected by variations in temperature and vertical velocity. Beam heating exceeds conductive heating at the heights where the spectral lines form, indicating that the electrons should contribute to the heating of the lower atmosphere and hence affect the line profiles. However, we find that it is difficult to determine specific signatures from the non-thermal electrons due to the complexity of the atmospheric response to the heating in combination with the relatively low energy output (~1e21 erg/s). Even so, our results contribute to understanding small-scale heating events in the solar atmosphere, and give further guidance to future observations.

While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is well known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition, as naive attempts to do this tend to fail. We introduce tools and definitions, in particular $\epsilon$-refinements and the $\epsilon$-erosion neighborhood of a module, to make sense of the question of how to build such a theory. Having built this foundation, we define the $\epsilon$-pruning of a module, which is a new invariant acting like a ``refined barcode'' that shows great promise to extract features from a module by approximately decomposing it. The main theorem of the paper can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of $2r$, where $r$ is the maximal pointwise dimension of one of the modules. Another interpretation is that the pruning of a module $M$ detects all summands appearing in modules in a neighborhood of $M$. Furthermore, we show that the factor $2r$ appearing in the theorem is close to optimal. We also define a distance that we conjecture to be a stable bottleneck-type distance, which has long been a missing piece in multipersistence theory. Finally, we discuss whether the main theorem can be improved for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the complexity of approximating the interleaving distance, and recent applications of homological algebra to multipersistence., Comment: 40 pages, 7 figures

We establish some properties of $\tau$-exceptional sequences for finite-dimensional algebras. In an earlier paper we established a bijection between the set of ordered support $\tau$-tilting modules and the set of complete signed $\tau$-exceptional sequences. We describe the action of the symmetric group on the latter induced by its natural action on the former. Similarly, we describe the effect on a $\tau$-exceptional sequence obtained by mutating the corresponding ordered support $\tau$-tilting module via a construction of Adachi-Iyama-Reiten., Comment: v2: only updated the supporting grant information

We have carried out a new photometric V,Rc study of 12 protoplanetary nebulae, objects in the short-lived transition between the AGB and PN phases of stellar evolution. These had been the subjects of an earlier study, using data from 1994-2007, that found that all 12 varied periodically, with pulsation periods in the range of ~38 to ~150 days. They are all carbon-rich, with F-G spectral types. We combined our new (2008-2018) data with publicly-available ASAS-SN data and determined new periods for their variability. The older and newer period values were compared to investigate evidence of period change, for which there is theoretical support that it might be detectable in a decade or two in some cases. Such a detection is challenging since the light curves are complicated, with multiple periods, changing amplitudes, and evidence of shocks. Nevertheless, we found one, and possibly two, such cases, which are associated with the higher temperature stars in the sample (7250 and 8000 K). These results are most consistent with the evolution of stars at the lower end of the mass range of carbon stars, ~1.5-2 M(sun). Several of the stars show longer-term trends of increasing (six cases) or decreasing (one case) brightness, which we think most likely due to changes in the circumstellar dust opacity. There is one case of a possible ~1.8 yr period in addition to the shorter pulsation. This is interpreted as possible evidence of an orbiting companion., Comment: 26 pages, 9 figures. Accepted September 16, 2022 for publication in the ApJ

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas., Comment: 33 pages, 4 figures; v2: add details, fix typos and minor issues, improve exposition, add conjecture 5.30; v3: minor improvements

We study the interaction of an electron/hole with inhomogeneous magnetic field in the presence of a screw dislocation. We consider the internal structure of the defect, i.e., we consider the core that gives rise to a finite size to the defect. In addition, we assume that this core determines a forbidden region for the electron/hole. Then, we solve the Schr\"odinger equation and show that an Aharonov-Bohm-type effect arises from the influence of the topological defect and the missing magnetic flux on the eigenvalues of energy., Comment: 12 pages, no figure

In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped with a multiparameter filtration. Multiparameter persistence has become one of the most active areas of research within TDA, with exciting progress on several fronts. In this article, we introduce multiparameter persistence and survey some of this recent progress, with a focus on ideas likely to lead to practical applications in the near future., Comment: Corrected several typos and addressed minor issues. Added a subsection on relative homological algebra. To appear in the proceedings of ICRA 2020

Heating signatures from small-scale magnetic reconnection events in the solar atmosphere have proven to be difficult to detect through observations. Numerical models that reproduce flaring conditions are essential in the understanding of how nanoflares may act as a heating mechanism of the corona. We study the effects of non-thermal electrons in synthetic spectra from 1D hydrodynamic RADYN simulations of nanoflare heated loops to investigate the diagnostic potential of chromospheric emission from small-scale events. The Mg II h and k, Ca II H and K, Ca II 854.2 nm, H-alpha and H-beta chromospheric lines were synthesised from various RADYN models of coronal loops subject to electron beams of nanoflare energies. The contribution function to the line intensity was computed to better understand how the atmospheric response to the non-thermal electrons affects the formation of spectral lines and the detailed shape of their spectral profiles. The spectral line signatures arising from the electron beams highly depend on the density of the loop and the lower cutoff energy of the electrons. Low-energy (5 keV) electrons deposit their energy in the corona and transition region, producing strong plasma flows that cause both redshifts and blueshifts of the chromospheric spectra. Higher-energy (10 and 15 keV) electrons deposit their energy in the lower transition region and chromosphere, resulting in increased emission from local heating. Our results indicate that effects from small-scale events can be observed with ground-based telescopes, expanding the list of possible diagnostics for the presence and properties of nanoflares.

We show that reconstructing a curve in $\mathbb{R}^d$ for $d\geq 2$ from a $0.66$-sample is always possible using an algorithm similar to the classical NN-Crust algorithm. Previously, this was only known to be possible for $0.47$-samples in $\mathbb{R}^2$ and $\frac{1}{3}$-samples in $\mathbb{R}^d$ for $d\geq 3$. In addition, we show that there is not always a unique way to reconstruct a curve from a $0.72$-sample; this was previously only known for $1$-samples. We also extend this non-uniqueness result to hypersurfaces in all higher dimensions., Comment: 24 pages, 16 figures. Full version of paper to be published in SoCG proceedings

We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case., Comment: 28 pages, 8 figures. This version establishes the tightness of all the bounds shown in the conference paper for SoCG 2022

Motivated by its links to $\tau$-tilting theory, we introduce a generalization of cotorsion pairs in module categories. Such pairs are also linked to co-t-structures in corresponding triangulated categories, and to cotorsion pairs in certain extension closed (and hence extriangulated) subcategories, which we call two-term categories., Comment: 15 pages. Any comments are welcome

In the field of topological data analysis, persistence modules are used to express geometrical features of data sets. The matching distance $d_\mathcal{M}$ measures the difference between $2$-parameter persistence modules by taking the maximum bottleneck distance between $1$-parameter slices of the modules. The previous best algorithm to compute $d_\mathcal{M}$ exactly runs in $O(n^{8+\omega})$ time using $O(n^4)$ space, where $n$ is the number of generators and relations of the modules and $\omega$ is the matrix multiplication constant. We improve significantly on this by describing an algorithm with expected running time $O(n^5 \log^3 n)$ and using $O(n^2)$ space. We first solve the decision problem $d_\mathcal{M}\leq \lambda$ for a constant $\lambda$ in $O(n^5\log n)$ time by traversing a line arrangement in the dual plane, where each point represents a slice. Then we lift the line arrangement to a plane arrangement in $\mathbb{R}^3$ whose vertices represent possible values for $d_\mathcal{M}$, and use a randomized incremental method to search through the vertices and find $d_\mathcal{M}$. The expected running time of this algorithm is $O((n^4+T(n))\log^2 n)$, where $T(n)$ is an upper bound for the complexity of deciding if $d_\mathcal{M}\leq \lambda$. Moreover, we show how to compute the matching distance using only linear space, to the price of a much worse time complexity.

Application of deep learning on histopathological whole slide images (WSIs) holds promise of improving diagnostic efficiency and reproducibility but is largely dependent on the ability to write computer code or purchase commercial solutions. We present a code-free pipeline utilizing free-to-use, open-source software (QuPath, DeepMIB, and FastPathology) for creating and deploying deep learning-based segmentation models for computational pathology. We demonstrate the pipeline on a use case of separating epithelium from stroma in colonic mucosa. A dataset of 251 annotated WSIs, comprising 140 hematoxylin-eosin (HE)-stained and 111 CD3 immunostained colon biopsy WSIs, were developed through active learning using the pipeline. On a hold-out test set of 36 HE and 21 CD3-stained WSIs a mean intersection over union score of 96.6% and 95.3% was achieved on epithelium segmentation. We demonstrate pathologist-level segmentation accuracy and clinical acceptable runtime performance and show that pathologists without programming experience can create near state-of-the-art segmentation solutions for histopathological WSIs using only free-to-use software. The study further demonstrates the strength of open-source solutions in its ability to create generalizable, open pipelines, of which trained models and predictions can seamlessly be exported in open formats and thereby used in external solutions. All scripts, trained models, a video tutorial, and the full dataset of 251 WSIs with ~31k epithelium annotations are made openly available at https://github.com/andreped/NoCodeSeg to accelerate research in the field., Comment: 18 pages, 4 figures, 2 tables

We investigate saddlepoint approximations applied to the score test statistic in genome-wide association studies with binary phenotypes. The inaccuracy in the normal approximation of the score test statistic increases with increasing sample imbalance and with decreasing minor allele count. Applying saddlepoint approximations to the score test statistic distribution greatly improve the accuracy, even far out in the tail of the distribution. By using exact results for an intercept model and binary covariate model, as well as simulations for models with nuisance parameters, we emphasize the need for continuity corrections in order to achieve valid $p$-values. The performance of the saddlepoint approximations is evaluated by overall and conditional type I error rate on simulated data. We investigate the methods further by using data from UK Biobank with skin and soft tissue infections as phenotype, using both common and rare variants. The analysis confirms that continuity correction is important particularly for rare variants, and that the normal approximation gives a highly inflated type I error rate for case imbalance., Comment: 15 pages in main manuscript and 7 pages in supplementary file

The persistent Betti numbers are used in topological data analysis to infer the scales at which topological features appear and disappear in the filtration of a topological space. Most commonly by means of the corresponding barcode or persistence diagram. While this approach to data science has been very successful, it suffers from sensitivity to outliers, and it does not allow for additional filtration parameters. Such parameters naturally appear when a cloud of data points comes together with additional measurements taken at the locations of the data. For these reasons, multiparameter persistent homology has recently received significant attention. In particular, the multicover and \v{C}ech bifiltration have been introduced to overcome the aforementioned shortcomings. In this work, we establish the strong consistency and asymptotic normality of the multiparameter persistent Betti numbers in growing domains. Our asymptotic results are established for a general framework encompassing both the marked \v{C}ech bifiltration, as well as the multicover bifiltration constructed on the null model of an independently marked Poisson point process. In a simulation study, we explain how the asymptotic normality can be used to derive tests for the goodness of fit. The statistical power of such tests is illustrated through different alternatives exhibiting more clustering, or more repulsion than the null model., Comment: 23 pages, 9 figures

The relative interlevel set cohomology (RISC) is an invariant of real-valued continuous functions closely related to the Mayer--Vietoris pyramid introduced by Carlsson, de Silva, and Morozov. As such, the relative interlevel set cohomology is a parametrization of the cohomology vector spaces of all open interlevel sets relative complements of closed interlevel sets. We provide a structure theorem, which applies to the RISC of real-valued continuous functions whose open interlevel sets have finite-dimensional cohomology in each degree. Moreover, we show this tameness assumption is in some sense equivalent to $q$-tameness as introduced by Chazal, de Silva, Glisse, and Oudot. Furthermore, we provide the notion of an interleaving for RISC and we show that it is stable in the sense that any space with two functions that are $\delta$-close induces a $\delta$-interleaving of the corresponding relative interlevel set cohomologies. Finally, we provide an elementary form of quantitative homotopy invariance for RISC., Comment: 40 pages + 10 pages appendix, 22 figures; this paper shares an appendix with an earlier version of arXiv:2007.01834; corrected the caption of figure 2.3, added missing assumption to corollary 3.15 (formerly corollary 3.13), added figure 3.2, added a corollary to theorem 3.5, added details to remark 4.1, added missing assumption to lemma A.15, strengthened lemma 2.3, several minor edits

In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $\Z$-linear combination of rank invariants of indicator modules supported on segments in the poset. We develop the theory behind these decompositions, both for the usual rank invariant and for its generalizations, showing under what conditions they exist and are unique. We also show that, like its unsigned counterpart, the signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive. To complete the picture, we show some experimental results that illustrate the contribution of the signed barcode in the exploration of multi-parameter persistence modules., Comment: The introduction and several other sections of the paper have been thoroughly rewritten, primarily for improved exposition. Minor issues have also been addressed. The paper has been accepted for publication in Foundations of Computational Mathematics

Let $\Lambda$ be a finite-dimensional algebra. A wide subcategory of $\mathsf{mod}\Lambda$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of $\mathsf{mod}\Lambda$ arising from $\tau$-tilting reduction are precisely the Serre subcategories of left finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau$-tilting reduction. This leads to a natural way to extend the definition of the "$\tau$-cluster morphism category" of $\Lambda$ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan-Marsh in the $\tau$-tilting finite case and by Igusa-Todorov in the hereditary case., Comment: v3: final version. v2: Removed Corollary 1.2a and added discussion of a counterexample as Remark 6.17, corrected errors in the proof of Theorem 1.3, and made other small changes to organization and exposition. 22 pages, 2 figures

Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two such generalizations $d_{\mathcal I}^p$ and $d_{\mathcal M}^p$, such that $d_{\mathcal I}^\infty$ equals the interleaving distance and $d_{\mathcal M}^\infty$ equals the matching distance. We show that on 1- or 2-parameter persistence modules over prime fields, $d_{\mathcal I}^p$ is the universal (i.e., largest) metric satisfying a natural stability property; this extends a stability theorem of Skraba and Turner for the $p$-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. We also show that $d_{\mathcal M}^p\leq d_{\mathcal I}^p$ for all $p\in [1,\infty]$, extending an observation of Landi in the $p=\infty$ case. We observe that on 2-parameter persistence modules, $d_{\mathcal M}^p$ can be efficiently approximated. In a forthcoming companion paper, we apply some of these results to study the stability of ($2$-parameter) multicover persistent homology., Comment: 49 pages. Rewrote beginning of introduction; other minor changes

This paper is based on a machine learning project at the Norwegian University of Science and Technology, fall 2020. The project was initiated with a literature review on the latest developments within time-series forecasting methods in the scientific community over the past five years. The paper summarizes the essential aspects of this research. Furthermore, in this paper, we introduce an LSTM cell's architecture, and explain how different components go together to alter the cell's memory and predict the output. Also, the paper provides the necessary formulas and foundations to calculate a forward iteration through an LSTM. Then, the paper refers to some practical applications and research that emphasize the strength and weaknesses of LSTMs, shown within the time-series domain and the natural language processing (NLP) domain. Finally, alternative statistical methods for time series predictions are highlighted, where the paper outline ARIMA and exponential smoothing. Nevertheless, as LSTMs can be viewed as a complex architecture, the paper assumes that the reader has some knowledge of essential machine learning aspects, such as the multi-layer perceptron, activation functions, overfitting, backpropagation, bias, over- and underfitting, and more.

This paper is a part of a student project in Machine Learning at the Norwegian University of Science and Technology. In this paper, a deep convolutional neural network with five convolutional layers and three fully-connected layers is presented to estimate the ages of individuals based on images. The model is in its entirety trained from scratch, where a combination of three different datasets is used as training data. These datasets are the APPA dataset, UTK dataset, and the IMDB dataset. The images were preprocessed using a proprietary face-recognition software. Our model is evaluated on both a held-out test set, and on the Adience benchmark. On the test set, our model achieves a categorical accuracy of 52%. On the Adience benchmark, our model proves inferior compared with other leading models, with an exact accuray of 30%, and an one-off accuracy of 46%. Furthermore, a script was created, allowing users to estimate their age directly using their web camera. The script, alongside all other code, is located in our GitHub repository: AgeNet.

We explore a scenario of the general relativity determined by the framework of the rainbow gravity with the purpose of searching for analogues of the Aharonov-Bohm effect. By focusing on the confinement of the Dirac field and the scalar field to a hard-wall confining potential in a modified background of the cosmic string spacetime, we examine the effects of the rainbow gravity and the topology of the cosmic string spacetime. Then, we compare each spectrum of energy with the cases where the rainbow gravity is absent., Comment: 18 pages, 2 figures

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so., Comment: Accepted in Algebras and Representation Theory. 44 pages. Proofs and exposition simplified. We thank the anonymous referees for their invaluable comments

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg--Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either., Comment: 30 pages + 15 pages appendix, 19 figures, LaTeX; moved an appendix to arXiv:2108.09298, generalization of results to restricted extended persistence diagrams, added discussion of relations to interleaving distances of sheaves and of Reeb graphs, correction of a sign mistake in the proof of lemma 3.2 with additional details in appendix D, updated references, several minor edits

The spin ice materials Ho2Ti2O7 and Dy2Ti2O7 are experimental and theoretical exemplars of highly frustrated magnetic materials. However, the effects of an applied uniaxial pressure are not well studied, and here we report magnetization measurements of Ho2Ti2O7 under uniaxial pressure applied in the [001], [111] and [110] crystalline directions. The basic features are captured by an extension of the dipolar spin ice model. We find a good match between our model and measurements with pressures applied along two of the three directions, and extend the framework to discuss the influence of crystal misalignment for the third direction. The parameters determined from the magnetization measurements reproduce neutron scattering measurements we perform under uniaxial pressure applied along the [110] crystalline direction. In the detailed analysis we include the recently verified susceptibility dependence of the demagnetizing factor. Our work demonstrates the application of a moderate applied pressure to modify the magnetic interaction parameters. The knowledge can be used to predict critical pressures needed to induce new phases and transitions in frustrated materials, and in the case of Ho2Ti2O7 we expect a transition to a ferromagnetic ground state for uniaxial pressures above 3.3 GPa., Comment: 13 pages, 14 figures

Observational and theoretical evidence suggest that beams of accelerated particles are produced in flaring events of all sizes in the solar atmosphere, from X-class flares to nanoflares. Current models of these types of particles in flaring loops assume an isolated 1D atmosphere. A more realistic environment for modelling accelerated particles can be provided by 3D radiative magnetohydrodynamics codes. Here, we present a simple model for particle acceleration and propagation in the context of a 3D simulation of the quiet solar atmosphere, spanning from the convection zone to the corona. We then examine the additional transport of energy introduced by the particle beams. The locations of particle acceleration associated with magnetic reconnection were identified by detecting changes in magnetic topology. At each location, the parameters of the accelerated particle distribution were estimated from local conditions. The particle distributions were then propagated along the magnetic field, and the energy deposition due to Coulomb collisions with the ambient plasma was computed. We find that particle beams originate in extended acceleration regions that are distributed across the corona. Upon reaching the transition region, they converge and produce strands of intense heating that penetrate the chromosphere. Within these strands, beam heating consistently dominates conductive heating below the bottom of the transition region. This indicates that particle beams qualitatively alter the energy transport even outside of active regions., Comment: Accepted for publication in A&A

The interleaving distance, although originally developed for persistent homology, has been generalized to measure the distance between functors modeled on many posets or even small categories. Existing theories require that such a poset have a superlinear family of translations or a similar structure. However, many posets of interest to topological data analysis, such as zig-zag posets and the face relation poset of a cell-complex, do not admit interesting translations, and consequently don't admit a nice theory of interleavings. In this paper we show how one can side-step this limitation by providing a general theory where one maps to a poset that does admit interesting translations, such as the lattice of down sets, and then defines interleavings relative to this map. Part of our theory includes a rigorous notion of discretization or "pixelization" of poset modules, which in turn we use for interleaving inference. We provide an approximation condition that in the setting of lattices gives rise to two possible pixelizations, both of which are guaranteed to be close in the interleaving distance. Finally, we conclude by considering interleaving inference for cosheaves over a metric space and give an explicit description of interleavings over a grid structure on Euclidean space., Comment: 50 pages, 2 figures

This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other hand, algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local characterization is key to the efficiency of our algorithms, and it generalizes previous conditions derived for the smaller class of block-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. By contrast, we show that general interval-decomposability does not admit such a local characterization, even when locality is understood in a broad sense. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work., Comment: This is the final version to be published in DCG. Changed the algorithm in Section 3 for computing the rank invariant, which was incorrect. The new algorithm reduces to computing 1-parameter vineyards along a family of paths in the grid. It was first proposed in the PhD thesis of D. Morozov

We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of a special type of persistence modules. We show that this matrix invertibility problem is NP-complete. We also give a slight improvement of the above reduction, showing that also the approximation of the interleaving distance is NP-hard for any approximation factor smaller than $3$. Additionally, we obtain corresponding hardness results for the case that the modules are indecomposable, and in the setting of one-sided stability. Furthermore, we show that checking for injections (resp. surjections) between persistence modules is NP-hard. In conjunction with earlier results from computational algebra this gives a complete characterization of the computational complexity of one-sided stability. Lastly, we show that it is in general NP-hard to approximate distances induced by noise systems within a factor of 2., Comment: 25 pages. Several expository improvements and minor corrections. Also added a section on noise systems

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete., Comment: Discussion related to the characteristic of the field added. Paper accepted to the 34th International Symposium on Computational Geometry

Non-relativistic quantum effects of the topology of the spacetime with the distortion of a vertical line into a vertical spiral on the harmonic oscillator are investigated. By searching for analytical solutions to the Schr\"odinger equation in this topological defect background, it is shown that the topology of the spacetime modifies the spectrum of energy of the harmonic oscillator. Besides, it is shown that there exists an Aharonov-Bohm-type effect for bound states., Comment: 5 pages

We consider an elastic medium with a disclination and investigate the topological effects on the interaction of a spinless electron with radial electric fields through the WKB (Wentzel, Kramers, Brillouin) approximation. We show how the centrifugal term of the radial equation must be modified due to the influence of the topological defect in order that the WKB approximation can be valid. Then, we search for bound states solutions from the interaction of a spinless electron with the electric field produced by this linear distribution of electric charges. In addition, we search for bound states solutions from the interaction of a spinless electron with radial electric field produced by uniform electric charge distribution inside a long non-conductor cylinder., Comment: 13 pages, no figures, Accepted for publication IJGMMP

It is investigated the effects of rotation on the scalar field in the spacetime with the distortion of a vertical line into a vertical spiral. By analysing the upper limit of the radial coordinate that stems from the effects of rotation and the topology of the defect, it is considered this upper limit of the radial coordinate as a boundary condition analogous to a hard-wall confining potential. Then, it is obtained a relativistic spectrum of energy in a particular case. In addition, it is analysed a relativistic analogue of the Aharonov-Bohm effect for bound states., Comment: 10 pages, no figure

Topological effects of a spiral dislocation on an electron are investigated when it is confined to a hard-wall confining potential. Besides, it is analysed the influence of the topology of the spiral dislocation on the interaction of the electron with a non-uniform radial electric field and a uniform axial magnetic field. It is shown that a discrete spectrum of energy can be obtained in all these cases. Moreover, it is shown that there is one case where an analogue of the Aharonov-Bohm effect for bound states is yielded by the topology of the spiral dislocation., Comment: 12 pages, no figure. arXiv admin note: text overlap with arXiv:1906.11000

The magnetically frustrated spin ice family of materials is host to numerous exotic phenomena such as magnetic monopole excitations and macroscopic residual entropy extending to low temperature. A finite-temperature ordering transition in the absence of applied fields has not been experimentally observed in the classical spin ice materials Dy2Ti2O7 and Ho2Ti2O7. Such a transition could be induced by the application of pressure, and in this work we consider the effects of uniaxial pressure on classical spin ice systems. Theoretically we find that the pressure induced ordering transition in Dy2Ti2O7 is strongly affected by the dipolar interaction. We also report measurements on the neutron structure factor of Ho2Ti2O7 under pressure, and compare the experimental results to the predictions of our theoretical model., Comment: 8 pages, 9 figures

It is analysed Aharonov-Bohm-type effects when a spinless quantum particle is in an elastic medium with the distortion of a vertical line into a vertical spiral. By confining the spinless particle to a cylindrical box, the analogue of the Aharonov-Bohm effect for bound states is observed due to influence of the topological defect on the allowed energies of the system. It corresponds to the shift in the angular momentum quantum number yielded by the effects of the topology of the distortion of a vertical line into a vertical spiral. In addition, it is analysed the effects of rotation. It is shown that the Aharonov-Bohm effect for bound states exists. Besides, there is an analogue of the coupling between the angular momentum and angular velocity of the rotating frame., Comment: 13 pages, no figure

By considering a spacetime with a spiral dislocation, we analyse the behaviour of the Dirac field subject to a hard-wall confining potential. In search of relativistic bound states solutions, we discuss the influence of the topology of the spiral dislocation spacetime on the energy levels. Further, we analyse the effects of rotation on the Dirac field in the spiral dislocation spacetime. We show that both rotation and the topology of the spacetime impose a restriction on the values of the radial coordinate. Thus, we analyse the effects of rotation and the topology of the spiral dislocation spacetime on the Dirac field subject to a hard-wall confining potential by searching for relativistic bound states solutions., Comment: 15 pages, no figure

The behaviour of the interaction of the induced electric dipole moment of an atom with a uniform magnetic field and a non-uniform electric field are investigated in a rotating reference frame. An interesting aspect of this interaction is that it gives rise to an analogue of a spinless particle subject to the doubly anharmonic oscillator. Then, it is shown that analytical solutions to the Schr\"odinger equation can be obtained. Another point raised is that the quantum effects on the induced electric dipole moment can be observed if the uniform magnetic field possesses a discrete set of values.

By applying the WKB (Wentzel, Kramers, Brillouin) approximation, we analyse the interaction of the electric quadrupole moment of a neutral particle with radial electric fields produced by a non-uniform electric charge densities. Then, we show particular cases of this interaction where a discrete spectrum of energy can be achieved for the s-waves of the neutral particle., Comment: 12 pages, no figure