TY - JOUR A1 - Chang, Der-Chen A1 - Khalil, Sara A1 - Schulze, Bert-Wolfgang T1 - Analysis on regular corner spaces JF - The journal of geometric analysis N2 - We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind. KW - Boutet de Monvel's calculus KW - Pseudo-differential operators KW - Singular cones KW - Mellin symbols with values in the edge calculus KW - Parametrices of elliptic operators KW - Kegel space Y1 - 2021 U6 - https://doi.org/10.1007/s12220-021-00614-3 SN - 1050-6926 SN - 1559-002X VL - 31 IS - 9 SP - 9199 EP - 9240 PB - Springer CY - New York ER - TY - JOUR A1 - Schindler, Daniel A1 - Moldenhawer, Ted A1 - Stange, Maike A1 - Lepro, Valentino A1 - Beta, Carsten A1 - Holschneider, Matthias A1 - Huisinga, Wilhelm T1 - Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows JF - PLoS Computational Biology : a new community journal N2 - Amoeboid cell motility is essential for a wide range of biological processes including wound healing, embryonic morphogenesis, and cancer metastasis. It relies on complex dynamical patterns of cell shape changes that pose long-standing challenges to mathematical modeling and raise a need for automated and reproducible approaches to extract quantitative morphological features from image sequences. Here, we introduce a theoretical framework and a computational method for obtaining smooth representations of the spatiotemporal contour dynamics from stacks of segmented microscopy images. Based on a Gaussian process regression we propose a one-parameter family of regularized contour flows that allows us to continuously track reference points (virtual markers) between successive cell contours. We use this approach to define a coordinate system on the moving cell boundary and to represent different local geometric quantities in this frame of reference. In particular, we introduce the local marker dispersion as a measure to identify localized membrane expansions and provide a fully automated way to extract the properties of such expansions, including their area and growth time. The methods are available as an open-source software package called AmoePy, a Python-based toolbox for analyzing amoeboid cell motility (based on time-lapse microscopy data), including a graphical user interface and detailed documentation. Due to the mathematical rigor of our framework, we envision it to be of use for the development of novel cell motility models. We mainly use experimental data of the social amoeba Dictyostelium discoideum to illustrate and validate our approach.
Author summary Amoeboid motion is a crawling-like cell migration that plays an important key role in multiple biological processes such as wound healing and cancer metastasis. This type of cell motility results from expanding and simultaneously contracting parts of the cell membrane. From fluorescence images, we obtain a sequence of points, representing the cell membrane, for each time step. By using regression analysis on these sequences, we derive smooth representations, so-called contours, of the membrane. Since the number of measurements is discrete and often limited, the question is raised of how to link consecutive contours with each other. In this work, we present a novel mathematical framework in which these links are described by regularized flows allowing a certain degree of concentration or stretching of neighboring reference points on the same contour. This stretching rate, the so-called local dispersion, is used to identify expansions and contractions of the cell membrane providing a fully automated way of extracting properties of these cell shape changes. We applied our methods to time-lapse microscopy data of the social amoeba Dictyostelium discoideum. Y1 - 2021 U6 - https://doi.org/10.1371/journal.pcbi.1009268 SN - 1553-734X SN - 1553-7358 VL - 17 IS - 8 PB - PLoS CY - San Fransisco ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarchanov, Nikolaj Nikolaevič T1 - An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n JF - Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija N2 - We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations. KW - Navier-Stokes equations KW - de Rham complex KW - open mapping theorem Y1 - 2021 U6 - https://doi.org/10.33048/semi.2021.18.108 SN - 1813-3304 VL - 18 IS - 2 SP - 1433 EP - 1466 PB - Institut Matematiki Imeni S. L. Soboleva CY - Novosibirsk ER - TY - JOUR A1 - Rodríguez Zuluaga, Juan A1 - Stolle, Claudia A1 - Yamazaki, Yosuke A1 - Xiong, Chao A1 - England, Scott L. T1 - A synoptic-scale wavelike structure in the nighttime equatorial ionization anomaly JF - Earth and Space Science : ESS N2 - Both ground- and satellite-based airglow imaging have significantly contributed to understanding the low-latitude ionosphere, especially the morphology and dynamics of the equatorial ionization anomaly (EIA). The NASA Global-scale Observations of the Limb and Disk (GOLD) mission focuses on far-ultraviolet airglow images from a geostationary orbit at 47.5 degrees W. This region is of particular interest at low magnetic latitudes because of the high magnetic declination (i.e., about -20 degrees) and proximity of the South Atlantic magnetic anomaly. In this study, we characterize an exciting feature of the nighttime EIA using GOLD observations from October 5, 2018 to June 30, 2020. It consists of a wavelike structure of a few thousand kilometers seen as poleward and equatorward displacements of the EIA-crests. Initial analyses show that the synoptic-scale structure is symmetric about the dip equator and appears nearly stationary with time over the night. In quasi-dipole coordinates, maxima poleward displacements of the EIA-crests are seen at about +/- 12 degrees latitude and around 20 and 60 degrees longitude (i.e., in geographic longitude at the dip equator, about 53 degrees W and 14 degrees W). The wavelike structure presents typical zonal wavelengths of about 6.7 x 10(3) km and 3.3 x 10(3) km. The structure's occurrence and wavelength are highly variable on a day-to-day basis with no apparent dependence on geomagnetic activity. In addition, a cluster or quasi-periodic wave train of equatorial plasma depletions (EPDs) is often detected within the synoptic-scale structure. We further outline the difference in observing these EPDs from FUV images and in situ measurements during a GOLD and Swarm mission conjunction. KW - equatorial ionization anomaly KW - equatorial ionosphere KW - equatorial plasma bubbles KW - wave structure KW - forcing from below Y1 - 2021 U6 - https://doi.org/10.1029/2020EA001529 SN - 2333-5084 VL - 8 IS - 2 PB - American Geophysical Union CY - Malden, Mass. ER - TY - JOUR A1 - Keller, Matthias A1 - Liu, Shiping A1 - Peyerimhoff, Norbert T1 - A note on eigenvalue bounds for non-compact manifolds JF - Mathematische Nachrichten N2 - In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general. KW - Cheeger inequality KW - eigenvalues KW - Laplacian KW - negative curvature KW - Riemannian manifold Y1 - 2021 U6 - https://doi.org/10.1002/mana.201900209 SN - 0025-584X SN - 1522-2616 VL - 294 IS - 6 SP - 1134 EP - 1139 PB - Wiley-VCH CY - Weinheim ER - TY - THES A1 - Zass, Alexander T1 - A multifaceted study of marked Gibbs point processes T1 - Facetten von markierten Gibbsschen Punktprozessen N2 - This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry. N2 - Diese Arbeit konzentriert sich auf die Untersuchung von markierten Gibbs-Punkt-Prozessen und stellt insbesondere einige Ergebnisse zu deren Existenz und Eindeutigkeit vor. Dabei werden Ideen und Techniken aus verschiedenen Bereichen der statistischen Mechanik verwendet: die Entropie-Methode aus der Theorie der großen Abweichungen, die Cluster-Expansion und die Kirkwood-Salsburg-Gleichungen, das Dobrushin-Kontraktionsprinzip und die Disagreement-Perkolation. Wir präsentieren zunächst ein Existenzergebnis für unendlich-volumige markierte Gibbs-Punkt-Prozesse. Genauer gesagt verwenden wir die sogenannte Entropie-Methode (und Werkzeuge der großen Abweichung), um markierte Gibbs-Punkt-Prozesse in R^d unter möglichst allgemeinen Annahmen zu konstruieren. Insbesondere gehören die zufälligen Markierungen zu einem allgemeinen normierten Raum und sind nicht beschränkt. Außerdem lassen wir Interaktionsfunktionale zu, die unbeschränkt sein können und deren Reichweite endlich, aber zufällig ist. Die Entropie-Methode beruht darauf, zu zeigen, dass eine Familie von endlich-volumigen Gibbs-Punkt-Prozessen zu sequentiell kompakten Entropie-Niveau-Mengen gehört, und daher dicht ist. Wir stellen dann unendlich-dimensionale Langevin-Diffusionen vor, die wir über eine Gibbssche Beschreibung in Wechselwirkung setzen. In dieser Umgebung sind wir in der Lage, das vorangehend vorgestellte allgemeine Ergebnis anzupassen, um die Existenz des zugehörigen unendlich-dimensionalen Maßes zu zeigen. Wir untersuchen auch seine Korrelationsfunktionen über Cluster-Expansions Techniken und erhalten die Eindeutigkeit des Gibbs-Prozesses für alle inversen Temperaturen β und Aktivitäten z unterhalb einer bestimmten Schwelle. Diese Methode beruht darauf, zunächst zu zeigen, dass die Korrelationsfunktionen des Prozesses eine so genannte Ruelle-Schranke erfüllen, um diese dann zur Lösung eines Fixpunktproblems in einem geeigneten Banach-Raum zu verwenden. Der Eindeutigkeitsbereich, den wir erhalten, wird dann aus den Modellparametern z und β definiert, für die ein solches Problem genau eine Lösung hat. Schließlich untersuchen wir die Frage nach der Eindeutigkeit von unendlich-volumigen Gibbs-Punkt-Prozessen auf R^d im unmarkierten Fall weiter. Im Zusammenhang mit repulsiven Wechselwirkungen basierend auf einer Hartkernkomponente stellen wir einen neuen Ansatz zur Eindeutigkeit vor, indem wir das diskrete Dobrushin-Kriterium im kontinuierlichen Rahmen anwenden. Wir legen zunächst einen Diskretisierungsparameter a>0 fest und untersuchen dann das Verhalten des Bereichs der Eindeutigkeit, wenn a gegen 0 geht. Mit dieser Technik sind wir in der Lage, explizite Schwellenwerte für die Parameter z und β zu erhalten, die wir dann mit bestehenden Ergebnissen aus den verschiedenen Methoden der Cluster-Expansion und der Disagreement-Perkolation vergleichen. In dieser Arbeit illustrieren wir unsere theoretischen Ergebnisse mit verschiedenen Beispielen sowohl aus der klassischen statistischen Mechanik als auch aus der stochastischen Geometrie. KW - marked Gibbs point processes KW - Langevin diffusions KW - Dobrushin criterion KW - Entropy method KW - Cluster expansion KW - Kirkwood--Salsburg equations KW - DLR equations KW - Markierte Gibbs-Punkt-Prozesse KW - Entropiemethode KW - Cluster-Expansion KW - DLR-Gleichungen KW - Dobrushin-Kriterium KW - Kirkwood-Salsburg-Gleichungen KW - Langevin-Diffusions Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-512775 ER - TY - JOUR A1 - Kempton, Mark A1 - Münch, Florentin A1 - Yau, Shing-Tung T1 - A homology vanishing theorem for graphs with positive curvature JF - Communications in analysis and geometry N2 - We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers. Y1 - 2021 UR - https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2021/0029/0006/CAG-2021-0029-0006-a005.pdf U6 - https://doi.org/10.4310/CAG.2021.v29.n6.a5 SN - 1019-8385 SN - 1944-9992 VL - 29 IS - 6 SP - 1449 EP - 1473 PB - International Press of Boston CY - Somerville ER - TY - JOUR A1 - Cvetković, Nada A1 - Conrad, Tim A1 - Lie, Han Cheng T1 - A convergent discretization method for transition path theory for diffusion processes JF - Multiscale modeling & simulation : a SIAM interdisciplinary journal N2 - Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential. KW - ergodic diffusion processes KW - transition paths KW - rare events KW - Monte Carlo KW - methods Y1 - 2021 U6 - https://doi.org/10.1137/20M1329354 SN - 1540-3459 SN - 1540-3467 VL - 19 IS - 1 SP - 242 EP - 266 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER -