@article{14770,
  abstract     = {We developed LIONESS, a technology that leverages improvements to optical super-resolution microscopy and prior information on sample structure via machine learning to overcome the limitations (in 3D-resolution, signal-to-noise ratio and light exposure) of optical microscopy of living biological specimens. LIONESS enables dense reconstruction of living brain tissue and morphodynamics visualization at the nanoscale.},
  author       = {Danzl, Johann G and Velicky, Philipp},
  issn         = {1548-7105},
  journal      = {Nature Methods},
  keywords     = {Cell Biology, Molecular Biology, Biochemistry, Biotechnology},
  number       = {8},
  pages        = {1141--1142},
  publisher    = {Springer Nature},
  title        = {{LIONESS enables 4D nanoscale reconstruction of living brain tissue}},
  doi          = {10.1038/s41592-023-01937-5},
  volume       = {20},
  year         = {2023},
}

@inproceedings{14771,
  abstract     = {Pruning—that is, setting a significant subset of the parameters of a neural network to zero—is one of the most popular methods of model compression. Yet, several recent works have raised the issue that pruning may induce or exacerbate bias in the output of the compressed model. Despite existing evidence for this phenomenon, the relationship between neural network pruning and induced bias is not well-understood. In this work, we systematically investigate and characterize this phenomenon in Convolutional Neural Networks for computer vision. First, we show that it is in fact possible to obtain highly-sparse models, e.g. with less than 10% remaining weights, which do not decrease in accuracy nor substantially increase in bias when compared to dense models. At the same time, we also find that, at higher sparsities, pruned models exhibit higher uncertainty in their outputs, as well as increased correlations, which we directly link to increased bias. We propose easy-to-use criteria which, based only on the uncompressed model, establish whether bias will increase with pruning, and identify the samples most susceptible to biased predictions post-compression. Our code can be found at https://github.com/IST-DASLab/pruned-vision-model-bias.},
  author       = {Iofinova, Eugenia B and Peste, Elena-Alexandra and Alistarh, Dan-Adrian},
  booktitle    = {2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  issn         = {2575-7075},
  location     = {Vancouver, BC, Canada},
  pages        = {24364--24373},
  publisher    = {IEEE},
  title        = {{Bias in pruned vision models: In-depth analysis and countermeasures}},
  doi          = {10.1109/cvpr52729.2023.02334},
  year         = {2023},
}

@article{14772,
  abstract     = {Many coupled evolution equations can be described via 2×2-block operator matrices of the form A=[ 
A	B
C	D
 ] in a product space X=X1×X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator A can be seen as a relatively bounded perturbation of its diagonal part with D(A)=D(A)×D(D) though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the H∞-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with A can be analyzed in maximal Lpt
-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.},
  author       = {Agresti, Antonio and Hussein, Amru},
  issn         = {0022-1236},
  journal      = {Journal of Functional Analysis},
  keywords     = {Analysis},
  number       = {11},
  publisher    = {Elsevier},
  title        = {{Maximal Lp-regularity and H∞-calculus for block operator matrices and applications}},
  doi          = {10.1016/j.jfa.2023.110146},
  volume       = {285},
  year         = {2023},
}

@article{14773,
  abstract     = {Through a combination of idealized simulations and real-world data, researchers are uncovering how internal feedbacks and large-scale motions influence cloud dynamics.},
  author       = {Muller, Caroline J and Abramian, Sophie},
  issn         = {1945-0699},
  journal      = {Physics Today},
  keywords     = {General Physics and Astronomy},
  number       = {5},
  publisher    = {AIP Publishing},
  title        = {{The cloud dynamics of convective storm systems}},
  doi          = {10.1063/pt.3.5234},
  volume       = {76},
  year         = {2023},
}

@article{14774,
  abstract     = {Morphogen gradients impart positional information to cells in a homogenous tissue field. Fgf8a, a highly conserved growth factor, has been proposed to act as a morphogen during zebrafish gastrulation. However, technical limitations have so far prevented direct visualization of the endogenous Fgf8a gradient and confirmation of its morphogenic activity. Here, we monitor Fgf8a propagation in the developing neural plate using a CRISPR/Cas9-mediated EGFP knock-in at the endogenous fgf8a locus. By combining sensitive imaging with single-molecule fluorescence correlation spectroscopy, we demonstrate that Fgf8a, which is produced at the embryonic margin, propagates by diffusion through the extracellular space and forms a graded distribution towards the animal pole. Overlaying the Fgf8a gradient curve with expression profiles of its downstream targets determines the precise input-output relationship of Fgf8a-mediated patterning. Manipulation of the extracellular Fgf8a levels alters the signaling outcome, thus establishing Fgf8a as a bona fide morphogen during zebrafish gastrulation. Furthermore, by hindering Fgf8a diffusion, we demonstrate that extracellular diffusion of the protein from the source is crucial for it to achieve its morphogenic potential.},
  author       = {Harish, Rohit K and Gupta, Mansi and Zöller, Daniela and Hartmann, Hella and Gheisari, Ali and Machate, Anja and Hans, Stefan and Brand, Michael},
  issn         = {1477-9129},
  journal      = {Development},
  keywords     = {Developmental Biology, Molecular Biology},
  number       = {19},
  publisher    = {The Company of Biologists},
  title        = {{Real-time monitoring of an endogenous Fgf8a gradient attests to its role as a morphogen during zebrafish gastrulation}},
  doi          = {10.1242/dev.201559},
  volume       = {150},
  year         = {2023},
}

@article{14775,
  abstract     = {We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.},
  author       = {Schnelli, Kevin and Xu, Yuanyuan},
  issn         = {1050-5164},
  journal      = {The Annals of Applied Probability},
  keywords     = {Statistics, Probability and Uncertainty, Statistics and Probability},
  number       = {1},
  pages        = {677--725},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices}},
  doi          = {10.1214/22-aap1826},
  volume       = {33},
  year         = {2023},
}

@article{14776,
  abstract     = {Soluble chaperones residing in the endoplasmic reticulum (ER) play vitally important roles in folding and quality control of newly synthesized proteins that transiently pass through the ER en route to their final destinations. These soluble residents of the ER are themselves endowed with an ER retrieval signal that enables the cell to bring the escaped residents back from the Golgi. Here, by using purified proteins, we showed that Nicotiana tabacum phytaspase, a plant aspartate-specific protease, introduces two breaks at the C-terminus of the N. tabacum ER resident calreticulin-3. These cleavages resulted in removal of either a dipeptide or a hexapeptide from the C-terminus of calreticulin-3 encompassing part or all of the ER retrieval signal. Consistently, expression of the calreticulin-3 derivative mimicking the phytaspase cleavage product in Nicotiana benthamiana cells demonstrated loss of the ER accumulation of the protein. Notably, upon its escape from the ER, calreticulin-3 was further processed by an unknown protease(s) to generate the free N-terminal (N) domain of calreticulin-3, which was ultimately secreted into the apoplast. Our study thus identified a specific proteolytic enzyme capable of precise detachment of the ER retrieval signal from a plant ER resident protein, with implications for the further fate of the escaped resident.},
  author       = {Teplova, Anastasiia and Pigidanov, Artemii A. and Serebryakova, Marina V. and Golyshev, Sergei A. and Galiullina, Raisa A. and Chichkova, Nina V. and Vartapetian, Andrey B.},
  issn         = {1422-0067},
  journal      = {International Journal of Molecular Sciences},
  keywords     = {Inorganic Chemistry, Organic Chemistry, Physical and Theoretical Chemistry, Computer Science Applications, Spectroscopy, Molecular Biology, General Medicine, Catalysis},
  number       = {22},
  publisher    = {MDPI},
  title        = {{Phytaspase Is capable of detaching the endoplasmic reticulum retrieval signal from tobacco calreticulin-3}},
  doi          = {10.3390/ijms242216527},
  volume       = {24},
  year         = {2023},
}

@article{14777,
  abstract     = {The effects of the partial V-substitution for Ag on the thermoelectric (TE) properties are investigated for a flexible semiconducting compound Ag2S0.55Se0.45. Density functional theory calculations predict that such a partial V-substitution constructively modifies the electronic structure near the bottom of the conduction band to improve the TE performance. The synthesized Ag1.97V0.03S0.55Se0.45 is found to possess a TE dimensionless figure-of-merit (ZT) of 0.71 at 350 K with maintaining its flexible nature. This ZT value is relatively high in comparison with those reported for flexible TE materials below 360 K. The increase in the ZT value is caused by the enhanced absolute value of the Seebeck coefficient with less significant variation in electrical resistivity. The high ZT value with the flexible nature naturally allows us to employ the Ag1.97V0.03S0.55Se0.45 as a component of flexible TE generators.},
  author       = {Sato, Kosuke and Singh, Saurabh and Yamazaki, Itsuki and Hirata, Keisuke and Ang, Artoni Kevin R. and Matsunami, Masaharu and Takeuchi, Tsunehiro},
  issn         = {2158-3226},
  journal      = {AIP Advances},
  keywords     = {General Physics and Astronomy},
  number       = {12},
  publisher    = {AIP Publishing},
  title        = {{Improvement of thermoelectric performance of flexible compound Ag2S0.55Se0.45 by means of partial V-substitution for Ag}},
  doi          = {10.1063/5.0171888},
  volume       = {13},
  year         = {2023},
}

@article{14778,
  abstract     = {We consider the almost-sure (a.s.) termination problem for probabilistic programs, which are a stochastic extension of classical imperative programs. Lexicographic ranking functions provide a sound and practical approach for termination of non-probabilistic programs, and their extension to probabilistic programs is achieved via lexicographic ranking supermartingales (LexRSMs). However, LexRSMs introduced in the previous work have a limitation that impedes their automation: all of their components have to be non-negative in all reachable states. This might result in a LexRSM not existing even for simple terminating programs. Our contributions are twofold. First, we introduce a generalization of LexRSMs that allows for some components to be negative. This standard feature of non-probabilistic termination proofs was hitherto not known to be sound in the probabilistic setting, as the soundness proof requires a careful analysis of the underlying stochastic process. Second, we present polynomial-time algorithms using our generalized LexRSMs for proving a.s. termination in broad classes of linear-arithmetic programs.},
  author       = {Chatterjee, Krishnendu and Kafshdar Goharshady, Ehsan and Novotný, Petr and Zárevúcky, Jiří and Zikelic, Dorde},
  issn         = {1433-299X},
  journal      = {Formal Aspects of Computing},
  keywords     = {Theoretical Computer Science, Software},
  number       = {2},
  publisher    = {Association for Computing Machinery},
  title        = {{On lexicographic proof rules for probabilistic termination}},
  doi          = {10.1145/3585391},
  volume       = {35},
  year         = {2023},
}

@article{14779,
  abstract     = {The presence of a developed boundary layer decouples a glacier's response from ambient conditions, suggesting that sensitivity to climate change is increased by glacier retreat. To test this hypothesis, we explore six years of distributed meteorological data on a small Swiss glacier in the period 2001–2022. Large glacier fragmentation has occurred since 2001 (−35% area change up to 2022) coinciding with notable frontal retreat, an observed switch from down‐glacier katabatic to up‐glacier valley winds and an increased sensitivity (ratio) of on‐glacier to off‐glacier temperature. As the glacier ceases to develop density‐driven katabatic winds, sensible heat fluxes on the glacier are increasingly determined by the conditions occurring outside the boundary layer of the glacier, sealing the glacier's demise as the climate continues to warm and experience an increased frequency of extreme summers.},
  author       = {Shaw, Thomas E. and Buri, Pascal and McCarthy, Michael and Miles, Evan S. and Ayala, Álvaro and Pellicciotti, Francesca},
  issn         = {1944-8007},
  journal      = {Geophysical Research Letters},
  keywords     = {General Earth and Planetary Sciences, Geophysics},
  number       = {11},
  publisher    = {American Geophysical Union},
  title        = {{The decaying near‐surface boundary layer of a retreating alpine glacier}},
  doi          = {10.1029/2023gl103043},
  volume       = {50},
  year         = {2023},
}

@article{14780,
  abstract     = {In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues λˆi and the generalized components (⟨v,uˆi⟩ for any deterministic vector v) of the outlier eigenvectors uˆi with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also hold near the so-called BBP transition and for degenerate spikes. On one hand, our results can be regarded as a refinement of the counterparts of [12] under additional regularity conditions. On the other hand, they can be viewed as an analog of [34] by replacing the random matrix with i.i.d. entries with Haar random matrix.},
  author       = {Ding, Xiucai and Ji, Hong Chang},
  issn         = {1879-209X},
  journal      = {Stochastic Processes and their Applications},
  keywords     = {Applied Mathematics, Modeling and Simulation, Statistics and Probability},
  pages        = {25--60},
  publisher    = {Elsevier},
  title        = {{Spiked multiplicative random matrices and principal components}},
  doi          = {10.1016/j.spa.2023.05.009},
  volume       = {163},
  year         = {2023},
}

@article{14781,
  abstract     = {Germ granules, condensates of phase-separated RNA and protein, are organelles that are essential for germline development in different organisms. The patterning of the granules and their relevance for germ cell fate are not fully understood. Combining three-dimensional in vivo structural and functional analyses, we study the dynamic spatial organization of molecules within zebrafish germ granules. We find that the localization of RNA molecules to the periphery of the granules, where ribosomes are localized, depends on translational activity at this location. In addition, we find that the vertebrate-specific Dead end (Dnd1) protein is essential for nanos3 RNA localization at the condensates’ periphery. Accordingly, in the absence of Dnd1, or when translation is inhibited, nanos3 RNA translocates into the granule interior, away from the ribosomes, a process that is correlated with the loss of germ cell fate. These findings highlight the relevance of sub-granule compartmentalization for post-transcriptional control and its importance for preserving germ cell totipotency.},
  author       = {Westerich, Kim Joana and Tarbashevich, Katsiaryna and Schick, Jan and Gupta, Antra and Zhu, Mingzhao and Hull, Kenneth and Romo, Daniel and Zeuschner, Dagmar and Goudarzi, Mohammad and Gross-Thebing, Theresa and Raz, Erez},
  issn         = {1534-5807},
  journal      = {Developmental Cell},
  keywords     = {Developmental Biology, Cell Biology, General Biochemistry, Genetics and Molecular Biology, Molecular Biology},
  number       = {17},
  pages        = {1578--1592.e5},
  publisher    = {Elsevier},
  title        = {{Spatial organization and function of RNA molecules within phase-separated condensates in zebrafish are controlled by Dnd1}},
  doi          = {10.1016/j.devcel.2023.06.009},
  volume       = {58},
  year         = {2023},
}

@article{14782,
  abstract     = {The actin cortex is a complex cytoskeletal machinery that drives and responds to changes in cell shape. It must generate or adapt to plasma membrane curvature to facilitate diverse functions such as cell division, migration, and phagocytosis. Due to the complex molecular makeup of the actin cortex, it remains unclear whether actin networks are inherently able to sense and generate membrane curvature, or whether they rely on their diverse binding partners to accomplish this. Here, we show that curvature sensing is an inherent capability of branched actin networks nucleated by Arp2/3 and VCA. We develop a robust method to encapsulate actin inside giant unilamellar vesicles (GUVs) and assemble an actin cortex at the inner surface of the GUV membrane. We show that actin forms a uniform and thin cortical layer when present at high concentration and distinct patches associated with negative membrane curvature at low concentration. Serendipitously, we find that the GUV production method also produces dumbbell-shaped GUVs, which we explain using mathematical modeling in terms of membrane hemifusion of nested GUVs. We find that branched actin networks preferentially assemble at the neck of the dumbbells, which possess a micrometer-range convex curvature comparable with the curvature of the actin patches found in spherical GUVs. Minimal branched actin networks can thus sense membrane curvature, which may help mammalian cells to robustly recruit actin to curved membranes to facilitate diverse cellular functions such as cytokinesis and migration.},
  author       = {Baldauf, Lucia and Frey, Felix F and Arribas Perez, Marcos and Idema, Timon and Koenderink, Gijsje H.},
  issn         = {0006-3495},
  journal      = {Biophysical Journal},
  keywords     = {Biophysics},
  number       = {11},
  pages        = {2311--2324},
  publisher    = {Elsevier},
  title        = {{Branched actin cortices reconstituted in vesicles sense membrane curvature}},
  doi          = {10.1016/j.bpj.2023.02.018},
  volume       = {122},
  year         = {2023},
}

@article{10145,
  abstract     = {We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.},
  author       = {Dello Schiavo, Lorenzo},
  issn         = {1572-929X},
  journal      = {Potential Analysis},
  pages        = {573--615},
  publisher    = {Springer Nature},
  title        = {{Ergodic decomposition of Dirichlet forms via direct integrals and applications}},
  doi          = {10.1007/s11118-021-09951-y},
  volume       = {58},
  year         = {2023},
}

@article{10173,
  abstract     = {We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R larger than 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.},
  author       = {Clozeau, Nicolas},
  issn         = {2194-0401},
  journal      = {Stochastics and Partial Differential Equations: Analysis and Computations},
  pages        = {1254–1378},
  publisher    = {Springer Nature},
  title        = {{Optimal decay of the parabolic semigroup in stochastic homogenization  for correlated coefficient fields}},
  doi          = {10.1007/s40072-022-00254-w},
  volume       = {11},
  year         = {2023},
}

@article{10405,
  abstract     = {We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1097-0312},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {5},
  pages        = {946--1034},
  publisher    = {Wiley},
  title        = {{Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices}},
  doi          = {10.1002/cpa.22028},
  volume       = {76},
  year         = {2023},
}

@article{10550,
  abstract     = {The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.},
  author       = {Fellner, Klemens and Fischer, Julian L and Kniely, Michael and Tang, Bao Quoc},
  issn         = {1432-1467},
  journal      = {Journal of Nonlinear Science},
  publisher    = {Springer Nature},
  title        = {{Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion}},
  doi          = {10.1007/s00332-023-09926-w},
  volume       = {33},
  year         = {2023},
}

@article{10551,
  abstract     = {The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N≫1. The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in N−1  (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.},
  author       = {Cornalba, Federico and Fischer, Julian L},
  issn         = {1432-0673},
  journal      = {Archive for Rational Mechanics and Analysis},
  number       = {5},
  publisher    = {Springer Nature},
  title        = {{The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles}},
  doi          = {10.1007/s00205-023-01903-7},
  volume       = {247},
  year         = {2023},
}

@article{10770,
  abstract     = {Mathematical models often aim to describe a complicated mechanism in a cohesive and simple manner. However, reaching perfect balance between being simple enough or overly simplistic is a challenging task. Frequently, game-theoretic models have an underlying assumption that players, whenever they choose to execute a specific action, do so perfectly. In fact, it is rare that action execution perfectly coincides with intentions of individuals, giving rise to behavioural mistakes. The concept of incompetence of players was suggested to address this issue in game-theoretic settings. Under the assumption of incompetence, players have non-zero probabilities of executing a different strategy from the one they chose, leading to stochastic outcomes of the interactions. In this article, we survey results related to the concept of incompetence in classic as well as evolutionary game theory and provide several new results. We also suggest future extensions of the model and argue why it is important to take into account behavioural mistakes when analysing interactions among players in both economic and biological settings.},
  author       = {Graham, Thomas and Kleshnina, Maria and Filar, Jerzy A.},
  issn         = {2153-0793},
  journal      = {Dynamic Games and Applications},
  pages        = {231--264},
  publisher    = {Springer Nature},
  title        = {{Where do mistakes lead? A survey of games with incompetent players}},
  doi          = {10.1007/s13235-022-00425-3},
  volume       = {13},
  year         = {2023},
}

@article{9651,
  abstract     = {We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞). Two separated nets are called ϕ-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most ϕ(R). We show that the spectrum of ϕ-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation, coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞. We further undertake a comparison of our notion of ϕ-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz equivalence.},
  author       = {Dymond, Michael and Kaluza, Vojtech},
  issn         = {1572-9168},
  journal      = {Geometriae Dedicata},
  publisher    = {Springer Nature},
  title        = {{Divergence of separated nets with respect to displacement equivalence}},
  doi          = {10.1007/s10711-023-00862-3},
  year         = {2023},
}

