@inproceedings{18599, abstract = {Hypernode logic can reason about the prefix relation on stutter-reduced finite traces through the stutter-reduced prefix predicate. We increase the expressiveness of hypernode logic in two ways. First, we split the stutter-reduced prefix predicate into an explicit stutter-reduction operator and the classical prefix predicate on words. This change gives hypernode logic the ability to combine synchronous and asynchronous reasoning by explicitly stating which parts of traces can stutter. Second, we allow the use of regular expressions in formulas to reason about the structure of traces. This change enables hypernode logic to describe a mixture of trace properties and hyperproperties. We show how to translate extended hypernode logic formulas into multi-track automata, which are automata that read multiple input words. Then we describe a fully online monitoring algorithm for monitoring k-safety hyperproperties specified in the logic. We have implemented the monitoring algorithm, and evaluated it on monitoring synchronous and asynchronous versions of observational determinism, and on checking the privacy preservation by compiler optimizations.}, author = {Chalupa, Marek and Henzinger, Thomas A and Oliveira da Costa, Ana}, booktitle = {Integrated Formal Methods}, isbn = {9783031765537}, issn = {1611-3349}, pages = {151--171}, publisher = {Springer Nature}, title = {{Monitoring extended hypernode logic}}, doi = {10.1007/978-3-031-76554-4_9}, volume = {15234}, year = {2024}, } @inproceedings{18600, abstract = {The analysis of formal models that include quantitative aspects such as timing or probabilistic choices is performed by quantitative verification tools. Broad and mature tool support is available for computing basic properties such as expected rewards on basic models such as Markov chains. Previous editions of QComp, the comparison of tools for the analysis of quantitative formal models, focused on this setting. Many application scenarios, however, require more advanced property types such as LTL and parameter synthesis queries as well as advanced models like stochastic games and partially observable MDPs. For these, tool support is in its infancy today. This paper presents the outcomes of QComp 2023: a survey of the state of the art in quantitative verification tool support for advanced property types and models. With tools ranging from first research prototypes to well-supported integrations into established toolsets, this report highlights today’s active areas and tomorrow’s challenges in tool-focused research for quantitative verification.}, author = {Andriushchenko, Roman and Bork, Alexander and Budde, Carlos E. and Češka, Milan and Grover, Kush and Hahn, Ernst Moritz and Hartmanns, Arnd and Israelsen, Bryant and Jansen, Nils and Jeppson, Joshua and Junges, Sebastian and Köhl, Maximilian A. and Könighofer, Bettina and Kretinsky, Jan and Meggendorfer, Tobias and Parker, David and Pranger, Stefan and Quatmann, Tim and Ruijters, Enno and Taylor, Landon and Volk, Matthias and Weininger, Maximilian and Zhang, Zhen}, booktitle = {TOOLympics Challenge 2023}, isbn = {9783031676949}, issn = {1611-3349}, pages = {90--146}, publisher = {Springer Nature}, title = {{Tools at the Frontiers of Quantitative Verification: QComp 2023 Competition Report}}, doi = {10.1007/978-3-031-67695-6_4}, volume = {14550}, year = {2024}, } @article{18602, abstract = {Semiconductor quantum dots (QDs) in planar germanium (Ge) heterostructures have emerged as front-runners for future hole-based quantum processors. Here, we present strong coupling between a hole charge qubit, defined in a double quantum dot (DQD) in planar Ge, and microwave photons in a high-impedance (Zr = 1.3 kΩ) resonator based on an array of superconducting quantum interference devices (SQUIDs). Our investigation reveals vacuum-Rabi splittings with coupling strengths up to g0/2π = 260 MHz, and a cooperativity of C ~ 100, dependent on DQD tuning. Furthermore, utilizing the frequency tunability of our resonator, we explore the quenched energy splitting associated with strong Coulomb correlation effects in Ge QDs. The observed enhanced coherence of the strongly correlated excited state signals the presence of distinct symmetries within related spin functions, serving as a precursor to the strong coupling between photons and spin-charge hybrid qubits in planar Ge. This work paves the way towards coherent quantum connections between remote hole qubits in planar Ge, required to scale up hole-based quantum processors.}, author = {De Palma, Franco and Oppliger, Fabian and Jang, Wonjin and Bosco, Stefano and Janik, Marian and Calcaterra, Stefano and Katsaros, Georgios and Isella, Giovanni and Loss, Daniel and Scarlino, Pasquale}, issn = {2041-1723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{Strong hole-photon coupling in planar Ge for probing charge degree and strongly correlated states}}, doi = {10.1038/s41467-024-54520-7}, volume = {15}, year = {2024}, } @article{18603, abstract = {It is widely believed that information storage in neuronal circuits involves nanoscopic structural changes at synapses, resulting in the formation of synaptic engrams. However, direct evidence for this hypothesis is lacking. To test this conjecture, we combined chemical potentiation, functional analysis by paired pre-postsynaptic recordings, and structural analysis by electron microscopy (EM) and freeze-fracture replica labeling (FRL) at the rodent hippocampal mossy fiber synapse, a key synapse in the trisynaptic circuit of the hippocampus. Biophysical analysis of synaptic transmission revealed that forskolin-induced chemical potentiation increased the readily releasable vesicle pool size and vesicular release probability by 146% and 49%, respectively. Structural analysis of mossy fiber synapses by EM and FRL demonstrated an increase in the number of vesicles close to the plasma membrane and the number of clusters of the priming protein Munc13-1, indicating an increase in the number of both docked and primed vesicles. Furthermore, FRL analysis revealed a significant reduction of the distance between Munc13-1 and CaV2.1 Ca2+ channels, suggesting reconfiguration of the channel-vesicle coupling nanotopography. Our results indicate that presynaptic plasticity is associated with structural reorganization of active zones. We propose that changes in potential nanoscopic organization at synaptic vesicle release sites may be correlates of learning and memory at a plastic central synapse.}, author = {Kim, Olena and Okamoto, Yuji and Kaufmann, Walter and Brose, Nils and Shigemoto, Ryuichi and Jonas, Peter M}, issn = {1545-7885}, journal = {PLoS Biology}, number = {11}, publisher = {Public Library of Science}, title = {{Presynaptic cAMP-PKA-mediated potentiation induces reconfiguration of synaptic vesicle pools and channel-vesicle coupling at hippocampal mossy fiber boutons}}, doi = {10.1371/journal.pbio.3002879}, volume = {22}, year = {2024}, } @article{18604, abstract = {A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a randomized FPT-time algorithm where the parameter is the number of popular faces.}, author = {De Nooijer, Phoebe and Terziadis, Soeren and Weinberger, Alexandra and Masárová, Zuzana and Mchedlidze, Tamara and Löffler, Maarten and Rote, Günter}, issn = {1526-1719}, journal = {Journal of Graph Algorithms and Applications}, number = {2}, pages = {47--82}, publisher = {Brown University}, title = {{Removing popular faces in curve arrangements}}, doi = {10.7155/jgaa.v28i2.2988}, volume = {28}, year = {2024}, } @article{18616, abstract = {By patterning an ultrathin layered structure with tiny wells, physicists have created and imaged peculiar states known as quantum scars — revealing behaviour that could be used to boost the performance of electronic devices.}, author = {Abanin, Dmitry and Serbyn, Maksym}, issn = {1476-4687}, journal = {Nature}, number = {8040}, pages = {825--826}, publisher = {Springer Nature}, title = {{Quantum scars make their mark in graphene}}, doi = {10.1038/d41586-024-03649-y}, volume = {635}, year = {2024}, } @article{18617, abstract = {Any complex-valued polynomial on (Rn)k decomposes into an algebraic combination of O(n)-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if n≥2k−1. We prove that the condition n≥2k−1 is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when n=2k−2 and n=2k−3. Formally, we study the kernel of a multiplication map ϕ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker ϕ in the restricted non-stable range k≤n<2k−1. In the full non-stable range n<2k−1, we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker ϕ.}, author = {Beďatš, Daniel}, issn = {0021-8693}, journal = {Journal of Algebra}, pages = {281--304}, publisher = {Elsevier}, title = {{Separation of variables for scalar-valued polynomials in the non-stable range}}, doi = {10.1016/j.jalgebra.2024.04.013}, volume = {651}, year = {2024}, } @article{18627, abstract = {In contrast with extended Bloch waves, a single particle can become spatially localized due to the so-called skin effect originating from non-Hermitian pumping. Here we show that in kinetically constrained many-body systems, the skin effect can instead manifest as dynamical amplification within the Fock space, beyond the intuitively expected and previously studied particle localization and clustering. We exemplify this non-Hermitian Fock skin effect in an asymmetric version of the PXP model and show that it gives rise to ergodicity-breaking eigenstates—the non-Hermitian analogs of quantum many-body scars. A distinguishing feature of these non-Hermitian scars is their enhanced robustness against external disorders. We propose an experimental realization of the non-Hermitian scar enhancement in a tilted Bose-Hubbard optical lattice with laser-induced loss. Additionally, we implement digital simulations of such scar enhancement on the IBM quantum processor. Our results show that the Fock skin effect provides a powerful tool for creating robust nonergodic states in generic open quantum systems.}, author = {Shen, Ruizhe and Qin, Fang and Desaules, Jean-Yves Marc and Papić, Zlatko and Lee, Ching Hua}, issn = {1079-7114}, journal = {Physical Review Letters}, number = {21}, publisher = {American Physical Society}, title = {{Enhanced many-body quantum scars from the non-hermitian fock skin effect}}, doi = {10.1103/PhysRevLett.133.216601}, volume = {133}, year = {2024}, } @article{18628, abstract = {Arctic tidewater glaciers are retreating, serving as key indicators of global warming. This study aims to assess how subglacial hydrology affects glacier front retreat by comparing two glacier–fjord models of the Hansbreen glacier: one incorporating a detailed subglacial hydrology model and another simplifying the subglacial discharge to a single channel centered in the flow line. We first validate the subglacial hydrology model by comparing its discharge channels with observations of plume activity. Simulations conducted from April to December 2010 revealed that the glacier front position aligns more closely with the observations in the coupled model than in the simplified version. Furthermore, the mass loss due to calving and submarine melting is greater in the coupled model, with the calving mass loss reaching 6 Mt by the end of the simulation compared to 4 Mt in the simplified model. These findings highlight the critical role of subglacial hydrology in predicting glacier dynamics and emphasize the importance of detailed modeling in understanding the responses of Arctic tidewater glaciers to climate change.}, author = {De Andrés, Eva and Muñoz Hermosilla, José M and Shahateet, Kaian and Otero, Jaime}, issn = {2306-5338}, journal = {Hydrology}, number = {11}, publisher = {MDPI}, title = {{The importance of solving Subglaciar hydrology in modeling glacier retreat: A case study of Hansbreen, Svalbard}}, doi = {10.3390/hydrology11110193}, volume = {11}, year = {2024}, } @article{18629, abstract = {We study a three-dimensional Gross-Pitaevskii equation that describes a static impurity in a dipolar Bose-Einstein condensate. Our focus is on the interplay between the shape of the impurity and the anisotropy of the medium manifested in the energy and the density of the system. Without external confinement, properties of the system are derived with basic analytical approaches. For a system in a harmonic trap, the model is investigated numerically, using the split-step Crank-Nicolson method. Our results demonstrate that the impurity self-energy is minimized when its shape more closely aligns with the anisotropic character of the bath; in particular a prolate deformed impurity aligned with the direction of the dipoles has the smallest self-energy for a repulsive impurity. Our work complements studies of impurities in Bose gases with zero-range interactions and paves the way for studies of dipolar polarons with a Gross-Pitaevskii equation.}, author = {Shukla, Neelam and Volosniev, Artem and Armstrong, Jeremy R.}, issn = {2469-9934}, journal = {Physical Review A}, number = {5}, publisher = {American Physical Society}, title = {{Anisotropic potential immersed in a dipolar Bose-Einstein condensate}}, doi = {10.1103/PhysRevA.110.053317}, volume = {110}, year = {2024}, } @article{18630, abstract = {Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.}, author = {Chatterjee, Krishnendu and Doyen, Laurent}, issn = {1860-5974}, journal = {Logical Methods in Computer Science}, number = {4}, pages = {11:1--11:34}, publisher = {EPI Sciences}, title = {{Stochastic processes with expected stopping time}}, doi = {10.46298/lmcs-20(4:11)2024}, volume = {20}, year = {2024}, } @article{18645, abstract = {Gene expression during natural and induced reprogramming is controlled by pioneer transcription factors that initiate transcription from closed chromatin. Nr5a2 is a key pioneer factor that regulates zygotic genome activation in totipotent embryos, pluripotency in embryonic stem cells and metabolism in adult tissues, but the mechanism of its pioneer activity remains poorly understood. Here, we present a cryo-electron microscopy structure of human NR5A2 bound to a nucleosome. The structure shows that the conserved carboxy-terminal extension (CTE) loop of the NR5A2 DNA-binding domain competes with a DNA minor groove anchor of the nucleosome and releases entry-exit site DNA. Mutational analysis showed that NR5A2 D159 of the CTE is dispensable for DNA binding but required for stable nucleosome association and persistent DNA ‘unwrapping’. These findings suggest that NR5A2 belongs to an emerging class of pioneer factors that can use DNA minor groove anchor competition to destabilize nucleosomes and facilitate gene expression during reprogramming.}, author = {Kobayashi, Wataru and Sappler, Anna H. and Bollschweiler, Daniel and Kümmecke, Maximilian and Basquin, Jérôme and Arslantas, Eda Nur and Ruangroengkulrith, Siwat and Hornberger, Renate and Duderstadt, Karl and Tachibana, Kikuë}, issn = {1545-9985}, journal = {Nature Structural & Molecular Biology}, pages = {757--766}, publisher = {Springer Nature}, title = {{Nucleosome-bound NR5A2 structure reveals pioneer factor mechanism by DNA minor groove anchor competition}}, doi = {10.1038/s41594-024-01239-0}, volume = {31}, year = {2024}, } @article{18651, abstract = {Embryo axis formation begins with the localized expression of biochemical signals, which organize cell movements and determine cell fate. A quail study finds that tissue contraction and resulting long-range changes in tissue tension restrict the area where these biochemical signals are expressed.}, author = {Hino, Naoya and Santos Fernandes Lasbarrères Camelo, Carolina and Heisenberg, Carl-Philipp J}, issn = {1879-0445}, journal = {Current Biology}, number = {24}, pages = {R1230--R1232}, publisher = {Elsevier}, title = {{Development: Turing mechanics}}, doi = {10.1016/j.cub.2024.10.065}, volume = {34}, year = {2024}, } @article{18652, abstract = {Over the last 70 years, information theory and coding has enabled communication technologies that have had an astounding impact on our lives. This is possible due to the match between encoding/decoding strategies and corresponding channel models. Traditional studies of channels have taken one of two extremes: Shannon-theoretic models are inherently average-case in which channel noise is governed by a memoryless stochastic process, whereas coding-theoretic (referred to as “Hamming”) models take a worst-case, adversarial, view of the noise. However, for several existing and emerging communication systems the Shannon/average-case view may be too optimistic, whereas the Hamming/worstcase view may be too pessimistic. This monograph takes up the challenge of studying adversarial channel models that lie between the Shannon and Hamming extremes.}, author = {Dey, Bikash Kumar and Jaggi, Sidharth and Langberg, Michael and Sarwate, Anand D. and Zhang, Yihan}, issn = {1567-2328}, journal = {Foundations and Trends in Communications and Information Theory}, number = {3-4}, pages = {300--588}, publisher = {Now Publishers}, title = {{Codes for adversaries: Between worst-case and average-case jamming}}, doi = {10.1561/0100000112}, volume = {21}, year = {2024}, } @article{18653, abstract = {Charge sensing is a sensitive technique for probing quantum devices, of particular importance for spin-qubit readout. To achieve good readout sensitivities, the proximity of the charge sensor to the device to be measured is a necessity. However, this proximity also means that the operation of the device affects, in turn, the sensor tuning and ultimately the readout sensitivity. We present an approach for compensating for this crosstalk effect allowing for the gate voltages of the measured device to be swept in a 1-V × 1-V window while maintaining a sensor configuration chosen by a Bayesian optimizer. Our algorithm will hopefully be a major contribution to the suite of fully automated solutions required for the operation of large quantum device architectures.}, author = {Hickie, Joseph and Van Straaten, Barnaby and Fedele, Federico and Jirovec, Daniel and Ballabio, Andrea and Chrastina, Daniel and Isella, Giovanni and Katsaros, Georgios and Ares, Natalia}, issn = {2331-7019}, journal = {Physical Review Applied}, number = {6}, publisher = {American Physical Society}, title = {{Automated long-range compensation of an rf quantum dot sensor}}, doi = {10.1103/PhysRevApplied.22.064026}, volume = {22}, year = {2024}, } @article{18654, abstract = {We compute the rotational anisotropy of the free energy of 𝛼−RuCl3 in an external magnetic field. This quantity, known as the magnetotropic susceptibility, 𝑘, relates to the second derivative of the free energy with respect to the angle of rotation. We have used approximation-free, auxiliary-field quantum Monte Carlo simulations for a realistic model of 𝛼−RuCl3 and optimized the path integral to alleviate the negative sign problem. This allows us to reach temperatures down to 30K—an energy scale below the dominant Kitaev coupling. We demonstrate that the magnetotropic spin susceptibility in this model of 𝛼−RuCl3 displays scaling behavior 𝑘=𝑇⁢𝑓⁡(𝐵/𝑇) at high temperatures. Once the uniform susceptibility departs from the Curie law (i.e., at the energy scale of the exchange interactions), it appears to transition to an emergent scalinglike behavior, characterized by a different function 𝑓 at lower temperatures, stemming from the locality of torque fluctuations. We observe a remarkable numerical match between experiment and simulations and we also find qualitative agreement with the pure Kitaev model. In comparison, for the XXZ Heisenberg Hamiltonian, the scaling 𝑘=𝑇⁢𝑓⁡(𝐵/𝑇) breaks down at a temperature scale where the uniform spin susceptibility deviates from the Curie law and never reemerges at low temperatures.}, author = {Sato, Toshihiro and Ramshaw, B. J. and Modic, Kimberly A and Assaad, Fakher F.}, issn = {2469-9969}, journal = {Physical Review B}, number = {20}, publisher = {American Physical Society}, title = {{Scale-invariant magnetic anisotropy in α-RuCl3: A quantum Monte Carlo study}}, doi = {10.1103/PhysRevB.110.L201114}, volume = {110}, year = {2024}, } @article{18655, abstract = {Let Qd be the d-dimensional binary hypercube. We say that P={v1,…,vk} is an increasing path of length k−1 in Qd, if for every i∈[k−1] the edge vivi+1 is obtained by switching some zero coordinate in vi to a one coordinate in vi+1. Form a random subgraph Qdp by retaining each edge in E(Qd) independently with probability p. We show that there is a phase transition with respect to the length of a longest increasing path around p=ed. Let α be a constant and let p=αd. When αe, whp there is a path of length d−2 in Qdp, and in fact, whether it is of length d−2,d−1, or d depends on whether the all-zero and all-one vertices percolate or not.}, author = {Anastos, Michael and Diskin, Sahar and Elboim, Dor and Krivelevich, Michael}, issn = {1083-589X}, journal = {Electronic Communications in Probability}, publisher = {Duke University Press}, title = {{Climbing up a random subgraph of the hypercube}}, doi = {10.1214/24-ECP639}, volume = {29}, year = {2024}, } @article{18656, abstract = {We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables and in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [10]) or the Hilbert-Schmidt norm (cf. [11]).}, author = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha}, issn = {1095-0753}, journal = {Advances in Theoretical and Mathematical Physics}, number = {6}, pages = {2025--2083}, publisher = {International Press}, title = {{Out-of-time-ordered correlators for Wigner matrices}}, doi = {10.4310/ATMP.241031013250}, volume = {28}, year = {2024}, } @phdthesis{18667, abstract = {Many chemical and physical properties of materials are determined by the material’s shape, for example the size of its pores and the width of its tunnels. This makes materials science a prime application area for geometrical and topological methods. Nevertheless many methods in topological data analysis have not been satisfyingly extended to the needs of materials science. This thesis provides new methods and new mathematical theorems targeted at those specific needs by answering four different research questions. While the motivation for each of the research questions arises from materials science, the methods are versatile and can be applied in different areas as well. The first research question is concerned with image data, for example a three-dimensional computed tomography (CT) scan of a material, like sand or stone. There are two commonly used topologies for digital images and depending on the application either of them might be required. However, software for computing the topological data analysis method persistence homology, usually supports only one of the two topologies. We answer the question how to compute persistent homology of an image with respect to one of the two topologies using software that is intended for the other topology. The second research question is concerned with image data as well, and asks how much of the topological information of an image is lost when the resolution is coarsened. As computer tomography scanners are more expensive the higher the resolution, it is an important question in materials science to know which resolution is enough to get satisfying persistent homology. We give theoretical bounds on the information loss based on different geometrical properties of the object to be scanned. In addition, we conduct experiments on sand and stone CT image data. The third research question is motivated by comparing crystalline materials efficiently. As the atoms within a crystal repeat periodically, crystalline materials are either modeled by unmanageable infinite periodic point sets, or by one of their fundamental domains, which is unstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with appropriate properties such that comparing the crystals can be eased by comparing the fingerprints instead. We define the density fingerprint and prove the necessary properties. The fourth research question is motivated by studying the hole-structure or connectedness, i.e. persistent homology or merge trees, of crystalline materials. A common way to deal with periodicity is to take a fundamental domain and identify opposite boundaries to form a torus. However, computing persistent homology or merge trees on that torus loses some of the information materials scientists are interested in and is additionally not stable under certain noise. We therefore decorate the merge tree stemming from the torus with additional information describing the density and growth rate of the periodic copies of a component within a growing spherical window. We prove all desired properties, like stability and efficient computability.}, author = {Heiss, Teresa}, isbn = {978-3-99078-052-7}, issn = {2663-337X}, keywords = {persistent homology, topological data analysis, periodic, crystalline materials, images, fingerprint}, pages = {111}, publisher = {Institute of Science and Technology Austria}, title = {{New methods for applying topological data analysis to materials science}}, doi = {10.15479/at:ista:18667}, year = {2024}, } @unpublished{18673, abstract = {Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes O((n+m)logn) time, in which n and m are the numbers of vertices and edges in the quotient complex, respectively. }, author = {Edelsbrunner, Herbert and Heiss, Teresa}, booktitle = {arXiv}, title = {{Merge trees of periodic filtrations}}, doi = {10.48550/arXiv.2408.16575}, year = {2024}, }