@article{ElinShoikhetTarkhanov2017,
  author    = {Elin, Mark and Shoikhet, David and Tarkhanov, Nikolai Nikolaevich},
  title     = {Analytic Semigroups of Holomorphic Mappings and Composition Operators},
  series = {Computational Methods and Function Theory},
  volume    = {18},
  journal   = {Computational Methods and Function Theory},
  number    = {2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1617-9447},
  doi       = {10.1007/s40315-017-0227-x},
  pages     = {269 -- 294},
  year      = {2017},
  abstract  = {In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators.},
  language  = {en}
}
@article{CarpentierKloppLoeffleretal.2018,
  author    = {Carpentier, Alexandra and Klopp, Olga and L{\"o}ffler, Matthias and Nickl, Richard},
  title     = {Adaptive confidence sets for matrix completion},
  series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  volume    = {24},
  journal   = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  number    = {4A},
  publisher = {International Statistical Institute},
  address   = {Voorburg},
  issn      = {1350-7265},
  doi       = {10.3150/17-BEJ933},
  pages     = {2429 -- 2460},
  year      = {2018},
  abstract  = {In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.},
  language  = {en}
}
@article{KoltaiLiePlonka2019,
  author    = {Koltai, Peter and Lie, Han Cheng and Plonka, Martin},
  title     = {Frechet differentiable drift dependence of Perron-Frobenius and Koopman operators for non-deterministic dynamics},
  series = {Nonlinearity},
  volume    = {32},
  journal   = {Nonlinearity},
  number    = {11},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0951-7715},
  doi       = {10.1088/1361-6544/ab1f2a},
  pages     = {4232 -- 4257},
  year      = {2019},
  abstract  = {We prove the Fr{\´e}chet differentiability with respect to the drift of Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron-Frobenius and Koopman operators.},
  language  = {en}
}
@article{Kollosche2017,
  author    = {Kollosche, David},
  title     = {Entdeckendes Lernen},
  series = {Journal f{\"u}r Mathematik-Didaktik},
  volume    = {38},
  journal   = {Journal f{\"u}r Mathematik-Didaktik},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {0173-5322},
  doi       = {10.1007/s13138-017-0116-x},
  pages     = {209 -- 237},
  year      = {2017},
  abstract  = {Trotz der nachweislichen Popularit{\"a}t des Entdeckenden Lernens in der deutschsprachigen Mathematikdidaktik finden sich aktuell keine kritischen Beitr{\"a}ge, die dazu beitragen k{\"o}nnten, dieses grundlegende Unterrichtskonzept zu hinterfragen und auszusch{\"a}rfen. In diesem Diskussionsbeitrag werden zun{\"a}chst die Theorie und einige Umsetzungsbeispiele des Entdeckenden Lernens herausgearbeitet, um aufzuzeigen, dass das Entdeckende Lernen einem vagen Sammelbegriff gleicht, unter dem oft fragw{\"u}rdige Unterrichtsumgebungen legitimiert werden. Anschließend werden an Hand erkenntnistheoretischer, lerntheoretischer, didaktischer und soziokultureller Betrachtungen Probleme des Entdeckenden Lernens im Mathematikunterricht und M{\"o}glichkeiten ihrer {\"U}berwindung thematisiert. Dabei zeigt sich, dass die Konzeption des Entdeckenden Lernens hinter dem aktuellen mathematikdidaktischen Erkenntnisstand zur{\"u}ckf{\"a}llt und Lehrer sowie Sch{\"u}ler mit unm{\"o}glichen Forderungen konfrontiert, dass lerntheoretische Vorteile des Entdeckenden Lernens oft nicht nachweisbar sind, dass die Idee des Entdeckens auf einem problematischen platonistischen Verst{\"a}ndnis von Erkenntnis beruht und dass Entdeckendes Lernen bildungsferne Sch{\"u}ler zu benachteiligen droht. Abschließend werden Forschungsdesiderata abgeleitet, deren Bearbeitung dazu beitragen k{\"o}nnte, die aufgezeigten Problemfelder zu {\"u}berwinden.},
  language  = {de}
}
@article{BandaraRosen2019,
  author    = {Bandara, Menaka Lashitha and Rosen, Andreas},
  title     = {Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions},
  series = {Communications in partial differential equations},
  volume    = {44},
  journal   = {Communications in partial differential equations},
  number    = {12},
  publisher = {Taylor \& Francis Group},
  address   = {Philadelphia},
  issn      = {0360-5302},
  doi       = {10.1080/03605302.2019.1611847},
  pages     = {1253 -- 1284},
  year      = {2019},
  abstract  = {On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.},
  language  = {en}
}
@article{RingelSomogyvariJalalietal.2019,
  author    = {Ringel, Lisa Maria and Somogyv{\´a}ri, M{\´a}rk and Jalali, Mohammadreza and Bayer, Peter},
  title     = {Comparison of hydraulic and tracer tomography for discrete fracture network inversion},
  series = {Geosciences},
  volume    = {9},
  journal   = {Geosciences},
  number    = {6},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {2076-3263},
  doi       = {10.3390/geosciences9060274},
  pages     = {17},
  year      = {2019},
  abstract  = {Fractures serve as highly conductive preferential flow paths for fluids in rocks, which are difficult to exactly reconstruct in numerical models. Especially, in low-conductive rocks, fractures are often the only pathways for advection of solutes and heat. The presented study compares the results from hydraulic and tracer tomography applied to invert a theoretical discrete fracture network (DFN) that is based on data from synthetic cross-well testing. For hydraulic tomography, pressure pulses in various injection intervals are induced and the pressure responses in the monitoring intervals of a nearby observation well are recorded. For tracer tomography, a conservative tracer is injected in different well levels and the depth-dependent breakthrough of the tracer is monitored. A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics. The used Metropolis-Hastings-Green algorithm is refined by the simultaneous estimation of the measurement error's variance, that is, the measurement noise. Based on the presented application to invert the two-dimensional cross-section between source and the receiver well, the hydraulic tomography reveals itself to be more suitable for reconstructing the original DFN. This is based on a probabilistic representation of the inverted results by means of fracture probabilities.},
  language  = {en}
}
@article{BeckusBellissardCornean2019,
  author    = {Beckus, Siegfried and Bellissard, Jean and Cornean, Horia},
  title     = {Holder Continuity of the Spectra for Aperiodic Hamiltonians},
  series = {Annales de l'Institut Henri Poincar{\´e}},
  volume    = {20},
  journal   = {Annales de l'Institut Henri Poincar{\´e}},
  number    = {11},
  publisher = {Springer},
  address   = {Cham},
  issn      = {1424-0637},
  doi       = {10.1007/s00023-019-00848-6},
  pages     = {3603 -- 3631},
  year      = {2019},
  abstract  = {We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Holder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.},
  language  = {en}
}
@article{BlanchardZadorozhnyi2019,
  author    = {Blanchard, Gilles and Zadorozhnyi, Oleksandr},
  title     = {Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods},
  series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  volume    = {25},
  journal   = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability},
  number    = {4B},
  publisher = {International Statistical Institute},
  address   = {Voorburg},
  issn      = {1350-7265},
  doi       = {10.3150/18-BEJ1095},
  pages     = {3421 -- 3458},
  year      = {2019},
  abstract  = {We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad family of spectral regularization methods for reproducing kernel decision rules, when trained on a sample coming from a tau-mixing process.},
  language  = {en}
}
@article{HoferTemmelHoudebert2018,
  author    = {Hofer-Temmel, Christoph and Houdebert, Pierre},
  title     = {Disagreement percolation for Gibbs ball models},
  series = {Stochastic processes and their application},
  volume    = {129},
  journal   = {Stochastic processes and their application},
  number    = {10},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0304-4149},
  doi       = {10.1016/j.spa.2018.11.003},
  pages     = {3922 -- 3940},
  year      = {2018},
  abstract  = {We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process. (C) 2018 Elsevier B.V. All rights reserved.},
  language  = {en}
}
@article{MeraStepanenkoTarkhanov2018,
  author    = {Mera, Azal and Stepanenko, Vitaly A. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Successive approximation for the inhomogeneous burgers equation},
  series = {Journal of Siberian Federal University : Mathematics \& Physics},
  volume    = {11},
  journal   = {Journal of Siberian Federal University : Mathematics \& Physics},
  number    = {4},
  publisher = {Siberian Federal University},
  address   = {Krasnoyarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2018-11-4-519-531},
  pages     = {519 -- 531},
  year      = {2018},
  abstract  = {The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term.},
  language  = {en}
}
@book{VanLeeuwenChengReich2015,
  author    = {Van Leeuwen, Peter Jan and Cheng, Yuan and Reich, Sebastian},
  title     = {Nonlinear data assimilation},
  series = {Frontiers in applied dynamical systems: reviews and tutorials ; 2},
  journal   = {Frontiers in applied dynamical systems: reviews and tutorials ; 2},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-319-18346-6},
  doi       = {10.1007/978-3-319-18347-3},
  pages     = {xii, 118},
  year      = {2015},
  abstract  = {This book contains two review articles on nonlinear data assimilation that deal with closely related topics but were written and can be read independently. Both contributions focus on so-called particle filters. The first contribution by Jan van Leeuwen focuses on the potential of proposal densities. It discusses the issues with present-day particle filters and explorers new ideas for proposal densities to solve them, converging to particle filters that work well in systems of any dimension, closing the contribution with a high-dimensional example. The second contribution by Cheng and Reich discusses a unified framework for ensemble-transform particle filters. This allows one to bridge successful ensemble Kalman filters with fully nonlinear particle filters, and allows a proper introduction of localization in particle filters, which has been lacking up to now.},
  language  = {en}
}
@misc{BlanchardScott2018,
  author    = {Blanchard, Gilles and Scott, Clayton},
  title     = {Corrigendum to: Classification with asymmetric label noise},
  series = {Electronic journal of statistics},
  volume    = {12},
  journal   = {Electronic journal of statistics},
  number    = {1},
  publisher = {Institute of Mathematical Statistics},
  address   = {Cleveland},
  issn      = {1935-7524},
  doi       = {10.1214/18-EJS1422},
  pages     = {1779 -- 1781},
  year      = {2018},
  abstract  = {We point out a flaw in Lemma 15 of [1]. We also indicate how the main results of that section are still valid using a modified argument.},
  language  = {en}
}
@misc{ShlapunovTarkhanov2017,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Golusin-Krylov formulas in complex analysis},
  series = {Complex variables and elliptic equations},
  volume    = {63},
  journal   = {Complex variables and elliptic equations},
  number    = {7-8},
  publisher = {Routledge},
  address   = {Abingdon},
  issn      = {1747-6933},
  doi       = {10.1080/17476933.2017.1395872},
  pages     = {1142 -- 1167},
  year      = {2017},
  abstract  = {This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.},
  language  = {en}
}
@article{MueckeBlanchard2018,
  author    = {M{\"u}cke, Nicole and Blanchard, Gilles},
  title     = {Parallelizing spectrally regularized kernel algorithms},
  series = {Journal of machine learning research},
  volume    = {19},
  journal   = {Journal of machine learning research},
  publisher = {Microtome Publishing},
  address   = {Cambridge, Mass.},
  issn      = {1532-4435},
  pages     = {29},
  year      = {2018},
  abstract  = {We consider a distributed learning approach in supervised learning for a large class of spectral regularization methods in an reproducing kernel Hilbert space (RKHS) framework. The data set of size n is partitioned into m = O (n(alpha)), alpha < 1/2, disjoint subsamples. On each subsample, some spectral regularization method (belonging to a large class, including in particular Kernel Ridge Regression, L-2-boosting and spectral cut-off) is applied. The regression function f is then estimated via simple averaging, leading to a substantial reduction in computation time. We show that minimax optimal rates of convergence are preserved if m grows sufficiently slowly (corresponding to an upper bound for alpha) as n -> infinity, depending on the smoothness assumptions on f and the intrinsic dimensionality. In spirit, the analysis relies on a classical bias/stochastic error analysis.},
  language  = {en}
}
@article{BlanchardHoffmannReiss2018,
  author    = {Blanchard, Gilles and Hoffmann, Marc and Reiss, Markus},
  title     = {Optimal adaptation for early stopping in statistical inverse problems},
  series = {SIAM/ASA Journal on Uncertainty Quantification},
  volume    = {6},
  journal   = {SIAM/ASA Journal on Uncertainty Quantification},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2166-2525},
  doi       = {10.1137/17M1154096},
  pages     = {1043 -- 1075},
  year      = {2018},
  abstract  = {For linear inverse problems Y = A mu + zeta, it is classical to recover the unknown signal mu by iterative regularization methods ((mu) over cap,(m) = 0,1, . . .) and halt at a data-dependent iteration tau using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error parallel to A((mu) over cap (()(tau)) - mu)parallel to(2) is controlled. In the context of statistical estimation with stochastic noise zeta, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared- error E[parallel to((mu) over cap (()(tau)) - mu)parallel to(2)]. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L-2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.},
  language  = {en}
}
@article{ShakiPinhasFischer2017,
  author    = {Shaki, Samuel and Pinhas, Michal and Fischer, Martin H.},
  title     = {Heuristics and biases in mental arithmetic},
  series = {Thinking \& Reasoning},
  volume    = {24},
  journal   = {Thinking \& Reasoning},
  number    = {2},
  publisher = {Routledge, Taylor \& Francis Group},
  address   = {Abingdon},
  issn      = {1354-6783},
  doi       = {10.1080/13546783.2017.1348987},
  pages     = {138 -- 156},
  year      = {2017},
  abstract  = {Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum (OM) effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems (e.g., 3+0, 3-0) but reverse OM with non-zero problems (e.g., 2+1, 4-1). In a third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving.},
  language  = {en}
}
@article{BlanchardHoffmannReiss2018,
  author    = {Blanchard, Gilles and Hoffmann, Marc and Reiss, Markus},
  title     = {Early stopping for statistical inverse problems via truncated SVD estimation},
  series = {Electronic journal of statistics},
  volume    = {12},
  journal   = {Electronic journal of statistics},
  number    = {2},
  publisher = {Institute of Mathematical Statistics},
  address   = {Cleveland},
  issn      = {1935-7524},
  doi       = {10.1214/18-EJS1482},
  pages     = {3204 -- 3231},
  year      = {2018},
  abstract  = {We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension D. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level (m) over cap is an element of {1, . . . , D} only based on the knowledge of the first (m) over cap singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.},
  language  = {en}
}
@article{BlanchardCarpentierGutzeit2018,
  author    = {Blanchard, Gilles and Carpentier, Alexandra and Gutzeit, Maurilio},
  title     = {Minimax Euclidean separation rates for testing convex hypotheses in R-d},
  series = {Electronic journal of statistics},
  volume    = {12},
  journal   = {Electronic journal of statistics},
  number    = {2},
  publisher = {Institute of Mathematical Statistics},
  address   = {Cleveland},
  issn      = {1935-7524},
  doi       = {10.1214/18-EJS1472},
  pages     = {3713 -- 3735},
  year      = {2018},
  abstract  = {We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset C of R-d. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension d and variance 1/n giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for C.},
  language  = {en}
}
@article{FriedrichKatzmannKrohmer2018,
  author    = {Friedrich, Tobias and Katzmann, Maximilian and Krohmer, Anton},
  title     = {Unbounded Discrepancy of Deterministic Random Walks on Grids},
  series = {SIAM journal on discrete mathematics},
  volume    = {32},
  journal   = {SIAM journal on discrete mathematics},
  number    = {4},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {0895-4801},
  doi       = {10.1137/17M1131088},
  pages     = {2441 -- 2452},
  year      = {2018},
  abstract  = {Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs called the rotor-router model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave in a remarkably similar way: Starting with the same initial configuration, the number of tokens in the rotor-router model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer [Combin. Probab. Comput., 15 (2006), pp. 815-822] showed that on Z(d), the single vertex discrepancy is only a constant c(d). Other authors also determined the precise value of c(d) for d = 1, 2. All of these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph Z(d). We show that this assumption is crucial by proving that, otherwise, the single vertex discrepancy can become arbitrarily large. For all dimensions d >= 1 and arbitrary discrepancies l >= 0, we construct configurations that reach a discrepancy of at least l.},
  language  = {en}
}
@article{BourneCushingLiuetal.2018,
  author    = {Bourne, D. P. and Cushing, D. and Liu, S. and M{\"u}nch, Florentin and Peyerimhoff, Norbert},
  title     = {Ollivier-Ricci idleness functions of graphs},
  series = {SIAM Journal on Discrete Mathematics},
  volume    = {32},
  journal   = {SIAM Journal on Discrete Mathematics},
  number    = {2},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {0895-4801},
  doi       = {10.1137/17M1134469},
  pages     = {1408 -- 1424},
  year      = {2018},
  abstract  = {We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most three linear parts, and at most two linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.},
  language  = {en}
}
@article{GomezHartung2018,
  author    = {Gomez, Christophe and Hartung, Niklas},
  title     = {Stochastic and deterministic models for the metastatic emission process},
  series = {Cancer Systems Biology},
  volume    = {1711},
  journal   = {Cancer Systems Biology},
  publisher = {Humana Press Inc.},
  address   = {Totowa},
  isbn      = {978-1-4939-7493-1},
  issn      = {1064-3745},
  doi       = {10.1007/978-1-4939-7493-1_10},
  pages     = {193 -- 224},
  year      = {2018},
  abstract  = {Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localized or metastatic disease. This chapter discusses mathematical modeling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: (1) a stochastic framework describing metastatic emission events at random times, formalized via Poisson processes, and (2) a deterministic framework describing the micrometastatic state through a size-structured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modeling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers.},
  language  = {en}
}
@article{Baer2019,
  author    = {B{\"a}r, Christian},
  title     = {The curl operator on odd-dimensional manifolds},
  series = {Journal of mathematical physics},
  volume    = {60},
  journal   = {Journal of mathematical physics},
  number    = {3},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0022-2488},
  doi       = {10.1063/1.5082528},
  pages     = {16},
  year      = {2019},
  abstract  = {We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions, its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres, and 3-dimensional spherical space forms. Published under license by AIP Publishing.},
  language  = {en}
}
@article{BeckerBeniniSchenkeletal.2019,
  author    = {Becker, Christian and Benini, Marco and Schenkel, Alexander and Szabo, Richard J.},
  title     = {Cheeger-Simons differential characters with compact support and Pontryagin duality},
  series = {Communications in analysis and geometry},
  volume    = {27},
  journal   = {Communications in analysis and geometry},
  number    = {7},
  publisher = {International Press of Boston},
  address   = {Somerville},
  issn      = {1019-8385},
  doi       = {10.4310/CAG.2019.v27.n7.a2},
  pages     = {1473 -- 1522},
  year      = {2019},
  abstract  = {By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125 (2003), 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology.},
  language  = {en}
}
@article{Reich2019,
  author    = {Reich, Sebastian},
  title     = {Data assimilation},
  series = {Acta numerica},
  volume    = {28},
  journal   = {Acta numerica},
  publisher = {Cambridge Univ. Press},
  address   = {New York},
  issn      = {0962-4929},
  doi       = {10.1017/S0962492919000011},
  pages     = {635 -- 711},
  year      = {2019},
  abstract  = {Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete- and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schr{\"o}dinger's boundary value problem for stochastic processes in particular.},
  language  = {en}
}
@article{MeraShlapunovTarkhanov2019,
  author    = {Mera, Azal and Shlapunov, Alexander A. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Navier-Stokes Equations for Elliptic Complexes},
  series = {Journal of Siberian Federal University. Mathematics \& Physics},
  volume    = {12},
  journal   = {Journal of Siberian Federal University. Mathematics \& Physics},
  number    = {1},
  publisher = {Sibirskij Federalʹnyj Universitet},
  address   = {Krasnojarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2019-12-1-3-27},
  pages     = {3 -- 27},
  year      = {2019},
  abstract  = {We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.},
  language  = {en}
}
@article{NueskenPavhotis2019,
  author    = {N{\"u}sken, Nikolas and Pavhotis, Grigorios A.},
  title     = {Constructing Sampling Schemes via Coupling},
  series = {SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association},
  volume    = {7},
  journal   = {SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association},
  number    = {1},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2166-2525},
  doi       = {10.1137/18M119896X},
  pages     = {324 -- 382},
  year      = {2019},
  abstract  = {In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincare inequality.},
  language  = {en}
}
@article{KatzSamuelsBlanchardScott2019,
  author    = {Katz-Samuels, Julian and Blanchard, Gilles and Scott, Clayton},
  title     = {Decontamination of Mutual Contamination Models},
  series = {Journal of machine learning research},
  volume    = {20},
  journal   = {Journal of machine learning research},
  publisher = {Microtome Publishing},
  address   = {Cambridge, Mass.},
  issn      = {1532-4435},
  pages     = {57},
  year      = {2019},
  abstract  = {Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees.},
  language  = {en}
}
@article{MalassTarkhanov2019,
  author    = {Malass, Ihsane and Tarkhanov, Nikolai Nikolaevich},
  title     = {The de Rham Cohomology through Hilbert Space Methods},
  series = {Journal of Siberian Federal University. Mathematics \& physics},
  volume    = {12},
  journal   = {Journal of Siberian Federal University. Mathematics \& physics},
  number    = {4},
  publisher = {Sibirskij Federalʹnyj Universitet},
  address   = {Krasnoyarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2019-12-4-455-465},
  pages     = {455 -- 465},
  year      = {2019},
  abstract  = {We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.},
  language  = {en}
}
@article{DombrowskyUndRoelly2019,
  author    = {Dombrowsky, Charlotte and Und, Myriam Fradon and Roelly, Sylvie},
  title     = {Packungen aus Kreisscheiben},
  series = {Elemente der Mathematik},
  volume    = {74},
  journal   = {Elemente der Mathematik},
  number    = {2},
  publisher = {EMS Publ.},
  address   = {Z{\"u}rich},
  issn      = {0013-6018},
  doi       = {10.4171/EM/381},
  pages     = {45 -- 62},
  year      = {2019},
  abstract  = {Der englische Seefahrer Sir Walter Raleigh fragte sich einst, wie er in seinem Schiffsladeraum moeglichst viele Kanonenkugeln stapeln koennte. Johannes Kepler entwickelte daraufhin 1611 eine Vermutung ueber die optimale Anordnung der Kugeln. Diese Vermutung sollte sich als eine der haertesten mathematischen Nuesse der Geschichte erweisen. Selbst in der Ebene sind dichteste Packungen kongruenter Kreise eine Herausforderung. 1892 und 1910 veroeffentlichte Axel Thue (kritisierte) Beweise, dass die hexagonale Kreispackung optimal sei. Erst 1940 lieferte Laszlo Fejes Toth schliesslich einen wasserdichten Beweis fuer diese Tatsache. Eine Variante des Problems verlangt, Packungen mit endlich vielen kongruenten Kugeln zu finden, die eine gewisse quadratische Energie minimieren: Diese spannende geometrische Aufgabe wurde 1967 von Toth gestellt. Sie ist auch heute noch nicht vollstaendig gelaest. In diesem Beitrag schlagen die Autorinnen eine originelle wahrscheinlichkeitstheoretische Methode vor, um in der Ebene N{\"a}herungen der L{\"o}sung zu konstruieren.},
  language  = {de}
}
@article{KleinRosenberger2018,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Tunneling for a class of difference operators},
  series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics},
  volume    = {19},
  journal   = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics},
  number    = {11},
  publisher = {Springer International Publishing},
  address   = {Cham},
  issn      = {1424-0637},
  doi       = {10.1007/s00023-018-0732-0},
  pages     = {3511 -- 3559},
  year      = {2018},
  abstract  = {We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a multi-well potential and ε is a small parameter. We derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two "wells" (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol h0(x,ξ) of Hε) connecting the two minima and the case where the minimal geodesics form an ℓ+1 dimensional manifold, ℓ≥1. These results on the tunneling problem are as sharp as the classical results for the Schr{\"o}dinger operator in Helffer and Sj{\"o}strand (Commun PDE 9:337-408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sj{\"o}strand [Analyse semi-classique pour l'{\´e}quation de Harper (avec application {\`a} l'{\´e}quation de Schr{\"o}dinger avec champ magn{\´e}tique), M{\´e}moires de la S.M.F., 2 series, tome 34, pp 1-113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincar{\´e} 60(2):147-187, 1994) to our discrete setting.},
  language  = {en}
}
@article{Paycha2018,
  author    = {Paycha, Sylvie},
  title     = {When the market wins over research and higher education},
  series = {Sustainable Futures for Higher Education : the Making of Knowledge Makers},
  volume    = {7},
  journal   = {Sustainable Futures for Higher Education : the Making of Knowledge Makers},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-319-96035-7},
  issn      = {2364-6799},
  doi       = {10.1007/978-3-319-96035-7_2},
  pages     = {23 -- 28},
  year      = {2018},
  abstract  = {In this chapter, an overview of systematic eradication of basic science foci in European universities in the last two decades is given. This happens under the slogan of optimisation of the university education to the needs and demands of the society. It is pointed out that reliance on "market demands" brings with it long-term deficiencies in the maintenance of basic and advanced knowledge construction in societies necessary for long-term future technological advances. University policies that claim improvement of higher education towards more immediate efficiency may end up with the opposite effect of affecting its quality and long term expected positive impact on society.},
  language  = {en}
}
@article{BachocBlanchardNeuvial2018,
  author    = {Bachoc, Francois and Blanchard, Gilles and Neuvial, Pierre},
  title     = {On the post selection inference constant under restricted isometry properties},
  series = {Electronic journal of statistics},
  volume    = {12},
  journal   = {Electronic journal of statistics},
  number    = {2},
  publisher = {Institute of Mathematical Statistics},
  address   = {Cleveland},
  issn      = {1935-7524},
  doi       = {10.1214/18-EJS1490},
  pages     = {3736 -- 3757},
  year      = {2018},
  abstract  = {Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound.},
  language  = {en}
}
@article{ConfortiPraRoelly2015,
  author    = {Conforti, Giovanni and Pra, Paolo Dai and Roelly, Sylvie},
  title     = {Reciprocal Class of Jump Processes},
  series = {Journal of theoretical probability},
  volume    = {30},
  journal   = {Journal of theoretical probability},
  publisher = {Springer},
  address   = {New York},
  issn      = {0894-9840},
  doi       = {10.1007/s10959-015-0655-3},
  pages     = {551 -- 580},
  year      = {2015},
  abstract  = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.},
  language  = {en}
}
@article{NueskenReichRozdeba2019,
  author    = {N{\"u}sken, Nikolas and Reich, Sebastian and Rozdeba, Paul J.},
  title     = {State and parameter estimation from observed signal increments},
  series = {Entropy : an international and interdisciplinary journal of entropy and information studies},
  volume    = {21},
  journal   = {Entropy : an international and interdisciplinary journal of entropy and information studies},
  number    = {5},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e21050505},
  pages     = {23},
  year      = {2019},
  abstract  = {The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.},
  language  = {en}
}
@article{MeraTarchanov2017,
  author    = {Mera, Azal Jaafar Musa and Tarchanov, Nikolaj Nikolaevič},
  title     = {The Neumann Problem after Spencer},
  series = {Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Matematika i fizika = Mathematics \& physics},
  volume    = {10},
  journal   = {Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Matematika i fizika = Mathematics \& physics},
  publisher = {Sibirskij Federalʹnyj Universitet},
  address   = {Krasnojarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2017-10-4-474-493},
  pages     = {474 -- 493},
  year      = {2017},
  abstract  = {When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.},
  language  = {en}
}
@article{SanchezWichtBaerenzung2020,
  author    = {Sanchez, Sabrina and Wicht, Johannes and B{\"a}renzung, Julien},
  title     = {Predictions of the geomagnetic secular variation based on the ensemble sequential assimilation of geomagnetic field models by dynamo simulations},
  series = {Earth, planets and space},
  volume    = {72},
  journal   = {Earth, planets and space},
  number    = {1},
  publisher = {Springer},
  address   = {New York},
  issn      = {1880-5981},
  doi       = {10.1186/s40623-020-01279-y},
  pages     = {20},
  year      = {2020},
  abstract  = {The IGRF offers an important incentive for testing algorithms predicting the Earth's magnetic field changes, known as secular variation (SV), in a 5-year range. Here, we present a SV candidate model for the 13th IGRF that stems from a sequential ensemble data assimilation approach (EnKF). The ensemble consists of a number of parallel-running 3D-dynamo simulations. The assimilated data are geomagnetic field snapshots covering the years 1840 to 2000 from the COV-OBS.x1 model and for 2001 to 2020 from the Kalmag model. A spectral covariance localization method, considering the couplings between spherical harmonics of the same equatorial symmetry and same azimuthal wave number, allows decreasing the ensemble size to about a 100 while maintaining the stability of the assimilation. The quality of 5-year predictions is tested for the past two decades. These tests show that the assimilation scheme is able to reconstruct the overall SV evolution. They also suggest that a better 5-year forecast is obtained keeping the SV constant compared to the dynamically evolving SV. However, the quality of the dynamical forecast steadily improves over the full assimilation window (180 years). We therefore propose the instantaneous SV estimate for 2020 from our assimilation as a candidate model for the IGRF-13. The ensemble approach provides uncertainty estimates, which closely match the residual differences with respect to the IGRF-13. Longer term predictions for the evolution of the main magnetic field features over a 50-year range are also presented. We observe the further decrease of the axial dipole at a mean rate of 8 nT/year as well as a deepening and broadening of the South Atlantic Anomaly. The magnetic dip poles are seen to approach an eccentric dipole configuration.},
  language  = {en}
}
@article{GarbunoInigoNueskenReich2020,
  author    = {Garbuno-Inigo, Alfredo and N{\"u}sken, Nikolas and Reich, Sebastian},
  title     = {Affine invariant interacting Langevin dynamics for Bayesian inference},
  series = {SIAM journal on applied dynamical systems},
  volume    = {19},
  journal   = {SIAM journal on applied dynamical systems},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1536-0040},
  doi       = {10.1137/19M1304891},
  pages     = {1633 -- 1658},
  year      = {2020},
  abstract  = {We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.},
  language  = {en}
}
@article{FroylandKoltaiStahn2020,
  author    = {Froyland, Gary and Koltai, Peter and Stahn, Martin},
  title     = {Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration},
  series = {SIAM journal on applied dynamical systems},
  volume    = {19},
  journal   = {SIAM journal on applied dynamical systems},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1536-0040},
  doi       = {10.1137/19M1261791},
  pages     = {1659 -- 1700},
  year      = {2020},
  abstract  = {Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations.},
  language  = {en}
}
@article{VasilievTarchanov2017,
  author    = {Vasiliev, Sergey B. and Tarchanov, Nikolaj Nikolaevič},
  title     = {Construction of series of perfect lattices by layer superposition},
  series = {Journal of Siberian Federal University : Mathematics \& physics},
  volume    = {10},
  journal   = {Journal of Siberian Federal University : Mathematics \& physics},
  number    = {3},
  publisher = {Sibirskij Federalʹnyj Universitet},
  address   = {Krasnojarsk},
  issn      = {1997-1397},
  doi       = {10.17516/1997-1397-2017-10-3-353-361},
  pages     = {353 -- 361},
  year      = {2017},
  abstract  = {We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.},
  language  = {en}
}
@misc{BandaraRosen2019,
  author    = {Bandara, Menaka Lashitha and Ros{\´e}n, Andreas},
  title     = {Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {758},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43407},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-434078},
  pages     = {1253 -- 1284},
  year      = {2019},
  abstract  = {On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.},
  language  = {en}
}
@phdthesis{Angwenyi2019,
  author    = {Angwenyi, David},
  title     = {Time-continuous state and parameter estimation with application to hyperbolic SPDEs},
  doi       = {10.25932/publishup-43654},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436542},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xi, 101},
  year      = {2019},
  abstract  = {Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state. The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling. The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations. The interconnection between It{\^o} integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes. We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter.},
  language  = {en}
}
@misc{MolahajlooPirhayati2013,
  author    = {Molahajloo, Shahla and Pirhayati, Mohammad},
  title     = {Traces of pseudo-differential operators on compact and Hausdorff groups},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  number    = {893},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43637},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436371},
  pages     = {361 -- 369},
  year      = {2013},
  abstract  = {We give a characterization of and a trace formula for trace class pseudo-differential operators on compact Hausdorff groups.},
  language  = {en}
}
@article{DereudreMazzonettoRoelly2017,
  author    = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie},
  title     = {Exact simulation of Brownian diffusions with drift admitting jumps},
  series = {SIAM journal on scientific computing},
  volume    = {39},
  journal   = {SIAM journal on scientific computing},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1064-8275},
  doi       = {10.1137/16M107699X},
  pages     = {A711 -- A740},
  year      = {2017},
  abstract  = {In this paper, using an algorithm based on the retrospective rejection sampling scheme introduced in [A. Beskos, O. Papaspiliopoulos, and G. O. Roberts,Methodol. Comput. Appl. Probab., 10 (2008), pp. 85-104] and [P. Etore and M. Martinez, ESAIM Probab.Stat., 18 (2014), pp. 686-702], we propose an exact simulation of a Brownian di ff usion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical di ffi culty due to the presence of t w o jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.},
  language  = {en}
}
@misc{DevchandNuytsWeingart2009,
  author    = {Devchand, Chandrashekar and Nuyts, Jean and Weingart, Gregor},
  title     = {Matryoshka of special democratic forms},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  number    = {841},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-42900},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-429002},
  pages     = {545 -- 562},
  year      = {2009},
  abstract  = {Special p-forms are forms which have components fµ1…µp equal to +1, -1 or 0 in some orthonormal basis. A p-form ϕ ∈ � pRd is called democratic if the set of nonzero components {ϕμ1...μp} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1, . . . , d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P = p and D = d. In particular, we display a remarkable nested structure of special forms including a U(3)-invariant 2-form in six dimensions, a G2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form O in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form.},
  language  = {en}
}
@article{Rastogi2020,
  author    = {Rastogi, Abhishake},
  title     = {Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems},
  series = {Communications on Pure and Applied Analysis},
  volume    = {19},
  journal   = {Communications on Pure and Applied Analysis},
  number    = {8},
  publisher = {American Institute of Mathematical Sciences},
  address   = {Springfield},
  issn      = {1534-0392},
  doi       = {10.3934/cpaa.2020183},
  pages     = {4111 -- 4126},
  year      = {2020},
  abstract  = {In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.},
  language  = {en}
}
@article{ChauhanFriedrichRothenberger2020,
  author    = {Chauhan, Ankit and Friedrich, Tobias and Rothenberger, Ralf},
  title     = {Greed is good for deterministic scale-free networks},
  series = {Algorithmica : an international journal in computer science},
  volume    = {82},
  journal   = {Algorithmica : an international journal in computer science},
  number    = {11},
  publisher = {Springer},
  address   = {New York},
  issn      = {0178-4617},
  doi       = {10.1007/s00453-020-00729-z},
  pages     = {3338 -- 3389},
  year      = {2020},
  abstract  = {Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306-1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical NP-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta)-approximation forMinimum Vertex Cover and Minimum Dominating Set, and a Omega(Delta)-approximation forMaximum Independent Set. We prove that the PLB-U property with beta>2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.},
  language  = {en}
}
@article{KleinRosenberger2021,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {The tunneling effect for Schr{\"o}dinger operators on a vector bundle},
  series = {Analysis and mathematical physics},
  volume    = {11},
  journal   = {Analysis and mathematical physics},
  number    = {2},
  publisher = {Springer International Publishing AG},
  address   = {Cham (ZG)},
  issn      = {1664-2368},
  doi       = {10.1007/s13324-021-00485-5},
  pages     = {35},
  year      = {2021},
  abstract  = {In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting.},
  language  = {en}
}
@article{Schlosser2022,
  author    = {Schlosser, Rainer},
  title     = {Heuristic mean-variance optimization in Markov decision processes using state-dependent risk aversion},
  series = {IMA journal of management mathematics / Institute of Mathematics and Its Applications},
  volume    = {33},
  journal   = {IMA journal of management mathematics / Institute of Mathematics and Its Applications},
  number    = {2},
  publisher = {Oxford Univ. Press},
  address   = {Oxford},
  issn      = {1471-678X},
  doi       = {10.1093/imaman/dpab009},
  pages     = {181 -- 199},
  year      = {2022},
  abstract  = {In dynamic decision problems, it is challenging to find the right balance between maximizing expected rewards and minimizing risks. In this paper, we consider NP-hard mean-variance (MV) optimization problems in Markov decision processes with a finite time horizon. We present a heuristic approach to solve MV problems, which is based on state-dependent risk aversion and efficient dynamic programming techniques. Our approach can also be applied to mean-semivariance (MSV) problems, which particularly focus on the downside risk. We demonstrate the applicability and the effectiveness of our heuristic for dynamic pricing applications. Using reproducible examples, we show that our approach outperforms existing state-of-the-art benchmark models for MV and MSV problems while also providing competitive runtimes. Further, compared to models based on constant risk levels, we find that state-dependent risk aversion allows to more effectively intervene in case sales processes deviate from their planned paths. Our concepts are domain independent, easy to implement and of low computational complexity.},
  language  = {en}
}
@article{Graeter2020,
  author    = {Gr{\"a}ter, Joachim},
  title     = {Free division rings of fractions of crossed products of groups with Conradian left-orders},
  series = {Forum mathematicum},
  volume    = {32},
  journal   = {Forum mathematicum},
  number    = {3},
  publisher = {De Gruyter},
  address   = {Berlin},
  issn      = {0933-7741},
  doi       = {10.1515/forum-2019-0264},
  pages     = {739 -- 772},
  year      = {2020},
  abstract  = {Let D be a division ring of fractions of a crossed product F[G, eta, alpha], where F is a skew field and G is a group with Conradian left-order <=. For D we introduce the notion of freeness with respect to <= and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F((G)) of all formal power series in G over F with respect to <=. From this we obtain that all division rings of fractions of F[G, eta, alpha] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F[G, eta, alpha] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G, eta, alpha] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field.},
  language  = {en}
}
@phdthesis{Busaman2006,
  author    = {Busaman, Saofee},
  title     = {Hyperequational theory for partial algebras},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-12048},
  school      = {Universit{\"a}t Potsdam},
  year      = {2006},
  abstract  = {Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n - SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V -iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions.},
  language  = {en}
}