@unpublished{BlanchardMuecke2016,
  author    = {Blanchard, Gilles and M{\"u}cke, Nicole},
  title     = {Optimal rates for regularization of statistical inverse learning problems},
  volume    = {5},
  number    = {5},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-89782},
  pages     = {36},
  year      = {2016},
  abstract  = {We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set.},
  language  = {en}
}
@unpublished{MeraTarkhanov2016,
  author    = {Mera, Azal and Tarkhanov, Nikolai Nikolaevich},
  title     = {The Neumann problem after Spencer},
  volume    = {5},
  number    = {6},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-90631},
  pages     = {21},
  year      = {2016},
  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}
}
@unpublished{MeraShlapunovTarkhanov2015,
  author    = {Mera, Azal and Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {Navier-Stokes equations for elliptic complexes},
  volume    = {4},
  number    = {12},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-85592},
  pages     = {27},
  year      = {2015},
  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}
}
@unpublished{Conforti2015,
  author    = {Conforti, Giovanni},
  title     = {Reciprocal classes of continuous time Markov Chains},
  volume    = {4},
  number    = {8},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78234},
  pages     = {198},
  year      = {2015},
  abstract  = {In this work we study reciprocal classes of Markov walks on graphs. Given a continuous time reference Markov chain on a graph, its reciprocal class is the set of all probability measures which can be represented as a mixture of the bridges of the reference walks. We characterize reciprocal classes with two different approaches. With the first approach we found it as the set of solutions to duality formulae on path space, where the differential operators have the interpretation of the addition of infinitesimal random loops to the paths of the canonical process. With the second approach we look at short time asymptotics of bridges. Both approaches allow an explicit computation of reciprocal characteristics, which are divided into two families, the loop characteristics and the arc characteristics. They are those specific functionals of the generator of the reference chain which determine its reciprocal class. We look at the specific examples such as Cayley graphs, the hypercube and planar graphs. Finally we establish the first concentration of measure results for the bridges of a continuous time Markov chain based on the reciprocal characteristics.},
  language  = {en}
}
@unpublished{GibaliShoikhetTarkhanov2015,
  author    = {Gibali, Aviv and Shoikhet, David and Tarkhanov, Nikolai Nikolaevich},
  title     = {On the convergence of continuous Newton method},
  volume    = {4},
  number    = {10},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-81537},
  pages     = {15},
  year      = {2015},
  abstract  = {In this paper we study the convergence of continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on a recent progress in the geometric theory of spirallike functions. We prove convergence theorems and illustrate them by numerical simulations.},
  language  = {en}
}
@phdthesis{Cheng2016,
  author    = {Cheng, Yuan},
  title     = {Recursive state estimation in dynamical systems},
  school      = {Universit{\"a}t Potsdam},
  pages     = {84},
  year      = {2016},
  language  = {en}
}
@phdthesis{Mera2017,
  author    = {Mera, Azal Jaafar Musa},
  title     = {The Navier-Stokes equations for elliptic quasicomplexes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-398495},
  school      = {Universit{\"a}t Potsdam},
  pages     = {101},
  year      = {2017},
  abstract  = {The classical Navier-Stokes equations of hydrodynamics are usually written in terms of vector analysis. More promising is the formulation of these equations in the language of differential forms of degree one. In this way the study of Navier-Stokes equations includes the analysis of the de Rham complex. In particular, the Hodge theory for the de Rham complex enables one to eliminate the pressure from the equations. The Navier-Stokes equations constitute a parabolic system with a nonlinear term which makes sense only for one-forms. A simpler model of dynamics of incompressible viscous fluid is given by Burgers' equation. This work is aimed at the study of invariant structure of the Navier-Stokes equations which is closely related to the algebraic structure of the de Rham complex at step 1. To this end we introduce Navier-Stokes equations related to any elliptic quasicomplex of first order differential operators. These equations are quite similar to the classical Navier-Stokes equations including generalised velocity and pressure vectors. Elimination of the pressure from the generalised Navier-Stokes equations gives a good motivation for the study of the Neumann problem after Spencer for elliptic quasicomplexes. Such a study is also included in the work.We start this work by discussion of Lam{\´e} equations within the context of elliptic quasicomplexes on compact manifolds with boundary. The non-stationary Lam{\´e} equations form a hyperbolic system. However, the study of the first mixed problem for them gives a good experience to attack the linearised Navier-Stokes equations. On this base we describe a class of non-linear perturbations of the Navier-Stokes equations, for which the solvability results still hold.},
  language  = {en}
}
@inproceedings{VallerianiRoellyKulik2017,
  author    = {Valleriani, Angelo and Roelly, Sylvie and Kulik, Alexei Michajlovič},
  title     = {Stochastic processes with applications in the natural sciences},
  series = {Lectures in pure and applied mathematics},
  booktitle = {Lectures in pure and applied mathematics},
  number    = {4},
  editor    = {Roelly, Sylvie and H{\"o}gele, Michael and Rafler, Mathias},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-414-2},
  issn      = {2199-4951},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-401802},
  pages     = {ix, 124},
  year      = {2017},
  abstract  = {The interdisciplinary workshop STOCHASTIC PROCESSES WITH APPLICATIONS IN THE NATURAL SCIENCES was held in Bogot{\´a}, at Universidad de los Andes from December 5 to December 9, 2016. It brought together researchers from Colombia, Germany, France, Italy, Ukraine, who communicated recent progress in the mathematical research related to stochastic processes with application in biophysics. The present volume collects three of the four courses held at this meeting by Angelo Valleriani, Sylvie Rœlly and Alexei Kulik. A particular aim of this collection is to inspire young scientists in setting up research goals within the wide scope of fields represented in this volume. Angelo Valleriani, PhD in high energy physics, is group leader of the team "Stochastic processes in complex and biological systems" from the Max-Planck-Institute of Colloids and Interfaces, Potsdam. Sylvie Rœlly, Docteur en Math{\´e}matiques, is the head of the chair of Probability at the University of Potsdam. Alexei Kulik, Doctor of Sciences, is a Leading researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences.},
  language  = {en}
}
@phdthesis{Muecke2017,
  author    = {M{\"u}cke, Nicole},
  title     = {Direct and inverse problems in machine learning},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-403479},
  school      = {Universit{\"a}t Potsdam},
  pages     = {159},
  year      = {2017},
  abstract  = {We analyze an inverse noisy regression model under random design with the aim of estimating the unknown target function based on a given set of data, drawn according to some unknown probability distribution. Our estimators are all constructed by kernel methods, which depend on a Reproducing Kernel Hilbert Space structure using spectral regularization methods. A first main result establishes upper and lower bounds for the rate of convergence under a given source condition assumption, restricting the class of admissible distributions. But since kernel methods scale poorly when massive datasets are involved, we study one example for saving computation time and memory requirements in more detail. We show that Parallelizing spectral algorithms also leads to minimax optimal rates of convergence provided the number of machines is chosen appropriately. We emphasize that so far all estimators depend on the assumed a-priori smoothness of the target function and on the eigenvalue decay of the kernel covariance operator, which are in general unknown. To obtain good purely data driven estimators constitutes the problem of adaptivity which we handle for the single machine problem via a version of the Lepskii principle.},
  language  = {en}
}
@unpublished{PolkovnikovTarkhanov2017,
  author    = {Polkovnikov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {A Riemann-Hilbert problem for the Moisil-Teodorescu system},
  volume    = {6},
  number    = {3},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-397036},
  pages     = {31},
  year      = {2017},
  abstract  = {In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations for a function u with values in R^3 subject to a nonhomogeneous condition (u,v)_x = u_0 on the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.},
  language  = {en}
}
@phdthesis{Jende2018,
  author    = {Jende, Alexander},
  title     = {On the characterization of particular orthogroups by disjunctions of identities},
  school      = {Universit{\"a}t Potsdam},
  pages     = {112},
  year      = {2018},
  abstract  = {In this thesis, we discuss the characterization of orthogroups by so-called disjunctions of identities. The orthogroups are a subclass of the class of completely regular semigroups, a generalization of the concept of a group. Thus there is for all elements of an orthogroup some kind of an inverse element such that both elements commute. Based on a fundamental result by A.H. Clifford, every completely regular semigroup is a semilattice of completely simple semigroups. This allows the description the gross structure of such semigroup. In particular every orthogroup is a semilattice of rectangular groups which are isomorphic to direct products of rectangular bands and groups. Semilattices of rectangular groups coming from various classes are characterized using the concept of an alternative variety, a generalization of the classical idea of a variety by Birkhoff. After starting with some fundamental definitions and results concerning semigroups, we introduce the concept of disjunctions of identities and summarize some necessary properties. In particular we present some disjunction of identities which is sufficient for a semigroup for being completely regular. Furthermore we derive from this identity some statements concerning Rees matrix semigroups, a possible representation of completely simple semigroups. A main result of this thesis is the general description of disjunctions of identities such that a completely regular semigroup satisfying the described identity is a semilattice of left groups (right groups / groups). In this case the completely regular semigroup is an orthogroup. Furthermore we define various classes of rectangular groups such that there is an exponent taken from a set of pairwise coprime positive integers. An important result is the characterization of the class of all semilattices of particular rectangular groups (taken from the classes defined before) using a set-theoretic minimal set of disjunctions of identities. Additionally we investigate semilattices of groups (so-called Clifford semigroups). For this purpose we consider abelian groups of particular exponents and prove some well-known results from the theory of Clifford semigroups in an alternative way applying the concept of disjunctions of identities. As a practical application of the results concerning semilattices of left zero semigroups and right zero semigroups we identify a particular transformation semigroup. For more detailed information about the product of two arbitrary elements of a semilattice of semigroups we introduce the concept of strong semilattices of semigroups. It is well-known that a semilattice of groups is a strong semilattice of groups. So we can characterize a strong semilattice of groups of particular pairwise coprime exponents by disjunctions of identities. Additionally we describe the class of all strong semilattices of left zero semigroups and right zero semigroups with the help of such kind of identity, and we relate this statement to the theory of normal bands. A possible extension of the already described semilattices of rectangular groups can be achieved by an auxiliary total order (in terms of chains of semigroups). To this end we present a corresponding characterization due to disjunctions of identities which is obviously minimal. A list of open questions which have arisen during the research for this thesis, but left crude, is attached.},
  language  = {en}
}
@phdthesis{Tinpun2019,
  author    = {Tinpun, Kittisak},
  title     = {Relative rank of infinite full transformation semigroups with restricted range},
  school      = {Universit{\"a}t Potsdam},
  year      = {2019},
  language  = {en}
}
@unpublished{LyTarkhanov2007,
  author    = {Ly, I. and Tarkhanov, Nikolai Nikolaevich},
  title     = {The cauchy problem for nonlinear elliptic equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30228},
  year      = {2007},
  abstract  = {This paper is devoted to investigation of the Cauchy problem for nonlinear elliptic equations with a small parameter.},
  language  = {en}
}
@phdthesis{Boehne2019,
  author    = {B{\"o}hne, Sebastian},
  title     = {Different degrees of formality},
  doi       = {10.25932/publishup-42379},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-423795},
  school      = {Universit{\"a}t Potsdam},
  pages     = {VI, 167},
  year      = {2019},
  abstract  = {In this thesis we introduce the concept of the degree of formality. It is directed against a dualistic point of view, which only distinguishes between formal and informal proofs. This dualistic attitude does not respect the differences between the argumentations classified as informal and it is unproductive because the individual potential of the respective argumentation styles cannot be appreciated and remains untapped. This thesis has two parts. In the first of them we analyse the concept of the degree of formality (including a discussion about the respective benefits for each degree) while in the second we demonstrate its usefulness in three case studies. In the first case study we will repair Haskell B. Curry's view of mathematics, which incidentally is of great importance in the first part of this thesis, in light of the different degrees of formality. In the second case study we delineate how awareness of the different degrees of formality can be used to help students to learn how to prove. Third, we will show how the advantages of proofs of different degrees of formality can be combined by the development of so called tactics having a medium degree of formality. Together the three case studies show that the degrees of formality provide a convincing solution to the problem of untapped potential.},
  language  = {en}
}
@article{KiyDehneBussler2015,
  author    = {Kiy, Alexander and Dehne, Julian and Bussler, Dirk},
  title     = {Aufbau einer Cloud-Speicherl{\"o}sung und Integration in bestehende IuK-Infrastrukturen am Beispiel ownCloud an der Universit{\"a}t Potsdam},
  series = {Cloudspeicher im Hochschuleinsatz 2015: Proceedings der Tagung "Cloudspeicher im Hochschuleinsatz" am 07. und 08. Mai 2015 am IT-Service-Center (tubIT) der Technischen Universit{\"a}t Berlin},
  journal   = {Cloudspeicher im Hochschuleinsatz 2015: Proceedings der Tagung "Cloudspeicher im Hochschuleinsatz" am 07. und 08. Mai 2015 am IT-Service-Center (tubIT) der Technischen Universit{\"a}t Berlin},
  publisher = {Universit{\"a}tsverlag der TU Berlin},
  address   = {Berlin},
  isbn      = {978-3-7983-2780-1},
  pages     = {61 -- 72},
  year      = {2015},
  abstract  = {In 2015 the second conference „Cloud Storage Deployment in Academics" took place. Interest regarding this issue was again high and topics established in 2014 like data security and scalability were complemented by new ones like federations or technical integration in existing infrastructures. This is caused by the advances in the establishment of cloud-based storage systems. This publication contains the contributions of the conference „Cloud Storage Deployment in Academics 2015", which took place in may 2015 at TU Berlin.},
  language  = {de}
}
@inproceedings{KiyStrelczukLucke2013,
  author    = {Kiy, Alexander and Strelczuk, Frederik and Lucke, Ulrike},
  title     = {Ein Plugin zur integrierten Literaturverwaltung in Moodle},
  series = {Ein Plugin zur integrierten Literaturverwaltung in Moodle},
  booktitle = {Ein Plugin zur integrierten Literaturverwaltung in Moodle},
  editor    = {Horbach, Matthias},
  publisher = {Gesellschaft f{\"u}r Informatik e.V.},
  address   = {Bonn},
  isbn      = {978-3-88579-614-5},
  pages     = {334 -- 345},
  year      = {2013},
  language  = {de}
}
@phdthesis{Muench2019,
  author    = {M{\"u}nch, Florentin},
  title     = {Discrete Ricci curvature, diameter bounds and rigidity},
  school      = {Universit{\"a}t Potsdam},
  pages     = {68},
  year      = {2019},
  language  = {en}
}
@article{HanischLudewig2022,
  author    = {Hanisch, Florian and Ludewig, Matthias},
  title     = {A rigorous construction of the supersymmetric path integral associated to a compact spin manifold},
  series = {Communications in mathematical physics},
  volume    = {391},
  journal   = {Communications in mathematical physics},
  number    = {3},
  publisher = {Springer},
  address   = {Berlin ; Heidelberg},
  issn      = {0010-3616},
  doi       = {10.1007/s00220-022-04336-7},
  pages     = {1209 -- 1239},
  year      = {2022},
  abstract  = {We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten.},
  language  = {en}
}
@article{SeeligRabeMalemShinitskietal.2020,
  author    = {Seelig, Stefan A. and Rabe, Maximilian Michael and Malem-Shinitski, Noa and Risse, Sarah and Reich, Sebastian and Engbert, Ralf},
  title     = {Bayesian parameter estimation for the SWIFT model of eye-movement control during reading},
  series = {Journal of mathematical psychology},
  volume    = {95},
  journal   = {Journal of mathematical psychology},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0022-2496},
  doi       = {10.1016/j.jmp.2019.102313},
  pages     = {32},
  year      = {2020},
  abstract  = {Process-oriented theories of cognition must be evaluated against time-ordered observations. Here we present a representative example for data assimilation of the SWIFT model, a dynamical model of the control of fixation positions and fixation durations during natural reading of single sentences. First, we develop and test an approximate likelihood function of the model, which is a combination of a spatial, pseudo-marginal likelihood and a temporal likelihood obtained by probability density approximation Second, we implement a Bayesian approach to parameter inference using an adaptive Markov chain Monte Carlo procedure. Our results indicate that model parameters can be estimated reliably for individual subjects. We conclude that approximative Bayesian inference represents a considerable step forward for computational models of eye-movement control, where modeling of individual data on the basis of process-based dynamic models has not been possible so far.},
  language  = {en}
}
@article{Pikovskij2021,
  author    = {Pikovskij, Arkadij},
  title     = {Chimeras on a social-type network},
  series = {Mathematical modelling of natural phenomena : MMNP},
  volume    = {16},
  journal   = {Mathematical modelling of natural phenomena : MMNP},
  publisher = {EDP Sciences},
  address   = {Les Ulis},
  issn      = {0973-5348},
  doi       = {10.1051/mmnp/2021012},
  pages     = {9},
  year      = {2021},
  abstract  = {We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.},
  language  = {en}
}
@book{PikovskijPoliti2016,
  author    = {Pikovskij, Arkadij and Politi, Antonio},
  title     = {Lyapunov Exponents},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  isbn      = {978-1-107-03042-8},
  publisher      = {Universit{\"a}t Potsdam},
  pages     = {XII, 285},
  year      = {2016},
  abstract  = {Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers.},
  language  = {en}
}
@misc{StraubePikovskij2011,
  author    = {Straube, Arthur V. and Pikovskij, Arkadij},
  title     = {Pattern formation induced by time-dependent advection},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  number    = {575},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-41314},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-413140},
  pages     = {138-147},
  year      = {2011},
  abstract  = {We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants.},
  language  = {en}
}
@article{DeOliveiraGomesHoegele2021,
  author    = {De Oliveira Gomes, Andr{\´e} and H{\"o}gele, Michael Anton},
  title     = {The Kramers problem for SDEs driven by small, accelerated L{\´e}vy noise with exponentially light jumps},
  series = {Stochastics and dynamics},
  volume    = {21},
  journal   = {Stochastics and dynamics},
  number    = {04},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4937},
  doi       = {10.1142/S0219493721500192},
  pages     = {44},
  year      = {2021},
  abstract  = {We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state 0, say, subject to a perturbation by a stochastic integral which is driven by an epsilon-small and (1/epsilon)-accelerated Levy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel, we solve the associated asymptotic first escape problem from the bounded neighborhood of 0 in the limit as epsilon -> 0 which is also known as the Kramers problem in the literature.},
  language  = {en}
}
@article{WiljesTong2020,
  author    = {Wiljes, Jana de and Tong, Xin T.},
  title     = {Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations},
  series = {Nonlinearity},
  volume    = {33},
  journal   = {Nonlinearity},
  number    = {9},
  publisher = {IOP Publ.},
  address   = {Bristol},
  issn      = {0951-7715},
  doi       = {10.1088/1361-6544/ab8d14},
  pages     = {4752 -- 4782},
  year      = {2020},
  abstract  = {Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.},
  language  = {en}
}
@phdthesis{Hanisch2011,
  author    = {Hanisch, Florian},
  title     = {Variational problems on supermanifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59757},
  school      = {Universit{\"a}t Potsdam},
  year      = {2011},
  abstract  = {In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields.},
  language  = {en}
}
@article{GarmendiaZambon2021,
  author    = {Garmendia, Alfonso and Zambon, Marco},
  title     = {Quotients of singular foliations and Lie 2-group actions},
  series = {Journal of noncommutative geometry},
  volume    = {15},
  journal   = {Journal of noncommutative geometry},
  number    = {4},
  publisher = {EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut f{\"u}r Mathematik, Technische Universit{\"a}t Berlin},
  address   = {Berlin},
  issn      = {1661-6952},
  doi       = {10.4171/JNCG/434},
  pages     = {1251 -- 1283},
  year      = {2021},
  abstract  = {Androulidakis-Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold (M, FM ) is the quotient of a foliated manifold (P, FP ) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.},
  language  = {en}
}
@article{SchickSeyedhosseini2021,
  author    = {Schick, Thomas and Seyedhosseini, Mehran},
  title     = {On an index theorem of Chang, Weinberger and Yu},
  series = {M{\"u}nster journal of mathematics},
  volume    = {14},
  journal   = {M{\"u}nster journal of mathematics},
  number    = {1},
  publisher = {WWU, Fachbereich Mathematik und Informatik},
  address   = {M{\"u}nster},
  issn      = {1867-5778},
  doi       = {10.17879/59019522628},
  pages     = {123 -- 154},
  year      = {2021},
  abstract  = {In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger and Yu (rather: a slight variation). To take the fundamental groups of the manifold and its boundary into account requires working with maximal C*-completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions. On the other hand, we introduce and propose a more conceptual and more geometric completion, which still has all the required functoriality.},
  language  = {en}
}
@article{TomovskiMetzlerGerhold2022,
  author    = {Tomovski, Živorad and Metzler, Ralf and Gerhold, Stefan},
  title     = {Fractional characteristic functions, and a fractional calculus approach for moments of random variables},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {25},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {4},
  publisher = {De Gruyter},
  address   = {Berlin ; Boston},
  issn      = {1314-2224},
  doi       = {10.1007/s13540-022-00047-x},
  pages     = {1307 -- 1323},
  year      = {2022},
  abstract  = {In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann-Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented.},
  language  = {en}
}
@article{PurintonBookhagen2020,
  author    = {Purinton, Benjamin and Bookhagen, Bodo},
  title     = {Multiband (X, C, L) radar amplitude analysis for a mixed sand- and gravel-bed river in the eastern Central Andes},
  series = {Remote sensing of environment : an interdisciplinary journal},
  volume    = {246},
  journal   = {Remote sensing of environment : an interdisciplinary journal},
  publisher = {Elsevier},
  address   = {New York},
  issn      = {0034-4257},
  doi       = {10.1016/j.rse.2020.111799},
  pages     = {16},
  year      = {2020},
  abstract  = {Synthetic Aperture Radar (SAR) amplitude measurements from spaceborne sensors are sensitive to surface roughness conditions near their radar wavelength. These backscatter signals are often exploited to assess the roughness of plowed agricultural fields and water surfaces, and less so to complex, heterogeneous geological surfaces. The bedload of mixed sand- and gravel-bed rivers can be considered a mixture of smooth (compacted sand) and rough (gravel) surfaces. Here, we assess backscatter gradients over a large high-mountain alluvial river in the eastern Central Andes with aerially exposed sand and gravel bedload using X-band TerraSAR-X/TanDEM-X, C-band Sentinel-1, and L-band ALOS-2 PALSAR-2 radar scenes. In a first step, we present theory and hypotheses regarding radar response to an alluvial channel bed. We test our hypotheses by comparing backscatter responses over vegetation-free endmember surfaces from inside and outside of the active channel-bed area. We then develop methods to extract smoothed backscatter gradients downstream along the channel using kernel density estimates. In a final step, the local variability of sand-dominated patches is analyzed using Fourier frequency analysis, by fitting stretched-exponential and power-law regression models to the 2-D power spectrum of backscatter amplitude. We find a large range in backscatter depending on the heterogeneity of contiguous smooth- and rough-patches of bedload material. The SAR amplitude signal responds primarily to the fraction of smooth-sand bedload, but is further modified by gravel elements. The sensitivity to gravel is more apparent in longer wavelength L-band radar, whereas C- and X-band is sensitive only to sand variability. Because the spatial extent of smooth sand patches in our study area is typically< 50 m, only higher resolution sensors (e.g., TerraSAR-X/TanDEM-X) are useful for power spectrum analysis. Our results show the potential for mapping sand-gravel transitions and local geomorphic complexity in alluvial rivers with aerially exposed bedload using SAR amplitude.},
  language  = {en}
}
@article{KwarikundaSchiefeleSsenyongaetal.2021,
  author    = {Kwarikunda, Diana and Schiefele, Ulrich and Ssenyonga, Joseph and Muwonge, Charles Magoba},
  title     = {Secondary school students' motivation profiles for physics learning},
  series = {African journal of research in mathematics, science and technology education : official journal of the Southern African Association for Research in Mathematics, Science and Technology Education},
  volume    = {25},
  journal   = {African journal of research in mathematics, science and technology education : official journal of the Southern African Association for Research in Mathematics, Science and Technology Education},
  number    = {2},
  publisher = {Routledge, Taylor \& Francis Group},
  address   = {Abingdon},
  issn      = {1028-8457},
  doi       = {10.1080/18117295.2021.1956720},
  pages     = {197 -- 210},
  year      = {2021},
  abstract  = {For efficient and effective pedagogical interventions to address Uganda's alarmingly poor performance in Physics, it is vital to understand students' motivation patterns for Physics learning. Latent profile analysis (LPA)-a person-centred approach-can be used to investigate these motivation patterns. Using a three-step approach to LPA, we sought to answer the following research questions: RQ1, which profiles of secondary school students exist with regards to their motivation for Physics learning; RQ2, are there differences in students' cognitive learning strategies in the identified profiles; and RQ3, does students' gender, attitudes, and individual interest predict membership in these profiles? The sample comprised 934 Grade 9 students from eight secondary schools in Uganda. Data were collected using standardised questionnaires. Six motivational profiles were identified: (i) low-quantity motivation profile (101 students; 10.8\%); (ii) moderate-quantity motivation profile (246 students; 26.3\%); (iii) high-quantity motivation profile (365 students; 39.1\%); (iv) primarily intrinsically motivated profile (60 students, 6.4\%); (v) mostly extrinsically motivated profile (88 students, 9.4\%); and (vi) grade-introjected profile (74 students, 7.9\%). Low-quantity and grade-introjected motivated students mostly used surface learning strategies whilst the high-quantity and primarily intrinsically motivated students used deep learning strategies. Lastly, unlike gender, individual interest and students' attitudes towards Physics learning predicted profile membership. Teachers should provide an interesting autonomous Physics classroom climate and give students clear instructions in self-reliant behaviours that promote intrinsic motivation.},
  language  = {en}
}
@unpublished{DereudreMazzonettoRoelly2016,
  author    = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie},
  title     = {Exact simulation of Brownian diffusions with drift admitting jumps},
  volume    = {5},
  number    = {7},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-91049},
  pages     = {25},
  year      = {2016},
  abstract  = {Using an algorithm based on a retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion 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 difficulty due to the presence of two 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{RoellyDereudre2004,
  author    = {Roelly, Sylvie and Dereudre, David},
  title     = {Propagation of Gibbsiannes for infinite-dimensional gradient Brownian diffusions},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6918},
  year      = {2004},
  abstract  = {We study the (strong-)Gibbsian character on R Z d of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian.},
  language  = {en}
}
@unpublished{DereudreRoelly2004,
  author    = {Dereudre, David and Roelly, Sylvie},
  title     = {Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51535},
  year      = {2004},
  abstract  = {We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian.},
  language  = {en}
}
@unpublished{DereudreMazzonettoRoelly2015,
  author    = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie},
  title     = {An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers},
  volume    = {4},
  number    = {9},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-80613},
  pages     = {23},
  year      = {2015},
  abstract  = {In this paper we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way.},
  language  = {en}
}
@unpublished{DereudreRoelly2014,
  author    = {Dereudre, David and Roelly, Sylvie},
  title     = {Path-dependent infinite-dimensional SDE with non-regular drift : an existence result},
  volume    = {3},
  number    = {11},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72084},
  pages     = {27},
  year      = {2014},
  abstract  = {We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.},
  language  = {en}
}
@misc{RoellyDereudre2004,
  author    = {Roelly, Sylvie and Dereudre, David},
  title     = {On Gibbsianness of infinite-dimensional diffusions},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6692},
  year      = {2004},
  abstract  = {The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. AMS Classifications: 60G15 , 60G60 , 60H10 , 60J60},
  language  = {en}
}
@article{DereudreRoelly2017,
  author    = {Dereudre, David and Roelly, Sylvie},
  title     = {Path-dependent infinite-dimensional SDE with non-regular drift},
  series = {Annales de l'Institut Henri Poincar{\´e} : B, Probability and statistics},
  volume    = {53},
  journal   = {Annales de l'Institut Henri Poincar{\´e} : B, Probability and statistics},
  number    = {2},
  publisher = {Inst. of Mathematical Statistics},
  address   = {Bethesda},
  issn      = {0246-0203},
  doi       = {10.1214/15-AIHP728},
  pages     = {641 -- 657},
  year      = {2017},
  abstract  = {We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither bounded or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy and a finite second moment. The originality of our method leads in the use of the specific entropy as a tightness tool and in the description of such infinite-dimensional stochastic process as solution of a variational problem on the path space. Our result clearly improves previous ones obtained for free dynamics with bounded drift.},
  language  = {en}
}
@phdthesis{Mazzonetto2016,
  author    = {Mazzonetto, Sara},
  title     = {On the exact simulation of (skew) Brownian diffusions with discontinuous drift},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-102399},
  school      = {Universit{\"a}t Potsdam},
  pages     = {ii, 100},
  year      = {2016},
  abstract  = {This thesis is focused on the study and the exact simulation of two classes of real-valued Brownian diffusions: multi-skew Brownian motions with constant drift and Brownian diffusions whose drift admits a finite number of jumps. The skew Brownian motion was introduced in the sixties by It{\^o} and McKean, who constructed it from the reflected Brownian motion, flipping its excursions from the origin with a given probability. Such a process behaves as the original one except at the point 0, which plays the role of a semipermeable barrier. More generally, a skew diffusion with several semipermeable barriers, called multi-skew diffusion, is a diffusion everywhere except when it reaches one of the barriers, where it is partially reflected with a probability depending on that particular barrier. Clearly, a multi-skew diffusion can be characterized either as solution of a stochastic differential equation involving weighted local times (these terms providing the semi-permeability) or by its infinitesimal generator as Markov process. In this thesis we first obtain a contour integral representation for the transition semigroup of the multiskew Brownian motion with constant drift, based on a fine analysis of its complex properties. Thanks to this representation we write explicitly the transition densities of the two-skew Brownian motion with constant drift as an infinite series involving, in particular, Gaussian functions and their tails. Then we propose a new useful application of a generalization of the known rejection sampling method. Recall that this basic algorithm allows to sample from a density as soon as one finds an - easy to sample - instrumental density verifying that the ratio between the goal and the instrumental densities is a bounded function. The generalized rejection sampling method allows to sample exactly from densities for which indeed only an approximation is known. The originality of the algorithm lies in the fact that one finally samples directly from the law without any approximation, except the machine's. As an application, we sample from the transition density of the two-skew Brownian motion with or without constant drift. The instrumental density is the transition density of the Brownian motion with constant drift, and we provide an useful uniform bound for the ratio of the densities. We also present numerical simulations to study the efficiency of the algorithm. The second aim of this thesis is to develop an exact simulation algorithm for a Brownian diffusion whose drift admits several jumps. In the literature, so far only the case of a continuous drift (resp. of a drift with one finite jump) was treated. The theoretical method we give allows to deal with any finite number of discontinuities. Then we focus on the case of two jumps, using the transition densities of the two-skew Brownian motion obtained before. Various examples are presented and the efficiency of our approach is discussed.},
  language  = {en}
}
@article{KemptonMuenchYau2021,
  author    = {Kempton, Mark and M{\"u}nch, Florentin and Yau, Shing-Tung},
  title     = {A homology vanishing theorem for graphs with positive curvature},
  series = {Communications in analysis and geometry},
  volume    = {29},
  journal   = {Communications in analysis and geometry},
  number    = {6},
  publisher = {International Press of Boston},
  address   = {Somerville},
  issn      = {1019-8385},
  doi       = {10.4310/CAG.2021.v29.n6.a5},
  pages     = {1449 -- 1473},
  year      = {2021},
  abstract  = {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.},
  language  = {en}
}
@misc{ButuzovNefedovReckeetal.2019,
  author    = {Butuzov, Valentin F. and Nefedov, N. N. and Recke, Lutz and Omel'chenko, Oleh},
  title     = {Partly dissipative system with multizonal initial and boundary layers},
  series = {Journal of Physics: Conference Series},
  volume    = {1205},
  journal   = {Journal of Physics: Conference Series},
  publisher = {IOP Publ.},
  address   = {Bristol},
  issn      = {1742-6588},
  doi       = {10.1088/1742-6596/1205/1/012009},
  pages     = {7},
  year      = {2019},
  abstract  = {For a singularly perturbed parabolic - ODE system we construct the asymptotic expansion in the small parameter in the case, when the degenerate equation has a double root. Such systems, which are called partly dissipative reaction-diffusion systems, are used to model various natural processes, including the signal transmission along axons, solid combustion and the kinetics of some chemical reactions. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. The multizonal initial and boundary layers behaviour was stated.},
  language  = {en}
}
@misc{Omel'chenko2019,
  author    = {Omel'chenko, Oleh},
  title     = {Travelling chimera states in systems of phase oscillators with asymmetric nonlocal coupling},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {2},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-51814},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-518141},
  pages     = {611 -- 642},
  year      = {2019},
  abstract  = {We study travelling chimera states in a ring of nonlocally coupled heterogeneous (with Lorentzian distribution of natural frequencies) phase oscillators. These states are coherence-incoherence patterns moving in the lateral direction because of the broken reflection symmetry of the coupling topology. To explain the results of direct numerical simulations we consider the continuum limit of the system. In this case travelling chimera states correspond to smooth travelling wave solutions of some integro-differential equation, called the Ott-Antonsen equation, which describes the long time coarse-grained dynamics of the oscillators. Using the Lyapunov-Schmidt reduction technique we suggest a numerical approach for the continuation of these travelling waves. Moreover, we perform their linear stability analysis and show that travelling chimera states can lose their stability via fold and Hopf bifurcations. Some of the Hopf bifurcations turn out to be supercritical resulting in the observation of modulated travelling chimera states.},
  language  = {en}
}
@article{Omel'chenko2022,
  author    = {Omel'chenko, Oleh},
  title     = {Mathematical framework for breathing chimera states},
  series = {Journal of nonlinear science},
  volume    = {32},
  journal   = {Journal of nonlinear science},
  number    = {2},
  publisher = {Springer},
  address   = {New York},
  issn      = {0938-8974},
  doi       = {10.1007/s00332-021-09779-1},
  pages     = {34},
  year      = {2022},
  abstract  = {About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique.},
  language  = {en}
}
@misc{ChechkinZaidLomholtetal.2013,
  author    = {Chechkin, Aleksei V. and Zaid, Irwin M. and Lomholt, Michael A. and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Bulk-mediated surface diffusion on a cylinder in the fast exchange limit},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  number    = {593},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-41548},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-415480},
  pages     = {114 -- 126},
  year      = {2013},
  abstract  = {In various biological systems and small scale technological applications particles transiently bind to a cylindrical surface. Upon unbinding the particles diffuse in the vicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise to an effective surface translation, for which we here derive and discuss the dynamic equations, including additional surface diffusion. We discuss the time evolution of the number of surface-bound particles, the effective surface mean squared displacement, and the surface propagator. In particular, we observe sub- and superdiffusive regimes. A plateau of the surface mean-squared displacement reflects a stalling of the surface diffusion at longer times. Finally, the corresponding first passage problem for the cylindrical geometry is analysed.},
  language  = {en}
}
@phdthesis{Rabe2024,
  author    = {Rabe, Maximilian Michael},
  title     = {Modeling the interaction of sentence processing and eye-movement control in reading},
  doi       = {10.25932/publishup-62279},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-622792},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xiii, 171},
  year      = {2024},
  abstract  = {The evaluation of process-oriented cognitive theories through time-ordered observations is crucial for the advancement of cognitive science. The findings presented herein integrate insights from research on eye-movement control and sentence comprehension during reading, addressing challenges in modeling time-ordered data, statistical inference, and interindividual variability. Using kernel density estimation and a pseudo-marginal likelihood for fixation durations and locations, a likelihood implementation of the SWIFT model of eye-movement control during reading (Engbert et al., Psychological Review, 112, 2005, pp. 777-813) is proposed. Within the broader framework of data assimilation, Bayesian parameter inference with adaptive Markov Chain Monte Carlo techniques is facilitated for reliable model fitting. Across the different studies, this framework has shown to enable reliable parameter recovery from simulated data and prediction of experimental summary statistics. Despite its complexity, SWIFT can be fitted within a principled Bayesian workflow, capturing interindividual differences and modeling experimental effects on reading across different geometrical alterations of text. Based on these advancements, the integrated dynamical model SEAM is proposed, which combines eye-movement control, a traditionally psychological research area, and post-lexical language processing in the form of cue-based memory retrieval (Lewis \& Vasishth, Cognitive Science, 29, 2005, pp. 375-419), typically the purview of psycholinguistics. This proof-of-concept integration marks a significant step forward in natural language comprehension during reading and suggests that the presented methodology can be useful to develop complex cognitive dynamical models that integrate processes at levels of perception, higher cognition, and (oculo-)motor control. These findings collectively advance process-oriented cognitive modeling and highlight the importance of Bayesian inference, individual differences, and interdisciplinary integration for a holistic understanding of reading processes. Implications for theory and methodology, including proposals for model comparison and hierarchical parameter inference, are briefly discussed.},
  language  = {en}
}
@article{MalemShinitskiOpperReichetal.2020,
  author    = {Malem-Shinitski, Noa and Opper, Manfred and Reich, Sebastian and Schwetlick, Lisa and Seelig, Stefan A. and Engbert, Ralf},
  title     = {A mathematical model of local and global attention in natural scene viewing},
  series = {PLoS Computational Biology : a new community journal},
  volume    = {16},
  journal   = {PLoS Computational Biology : a new community journal},
  number    = {12},
  publisher = {PLoS},
  address   = {San Fransisco},
  issn      = {1553-734X},
  doi       = {10.1371/journal.pcbi.1007880},
  pages     = {21},
  year      = {2020},
  abstract  = {Author summary <br /> Switching between local and global attention is a general strategy in human information processing. We investigate whether this strategy is a viable approach to model sequences of fixations generated by a human observer in a free viewing task with natural scenes. Variants of the basic model are used to predict the experimental data based on Bayesian inference. Results indicate a high predictive power for both aggregated data and individual differences across observers. The combination of a novel model with state-of-the-art Bayesian methods lends support to our two-state model using local and global internal attention states for controlling eye movements. <br /> Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.},
  language  = {en}
}
@phdthesis{Krejca2019,
  author    = {Krejca, Martin Stefan},
  title     = {Theoretical analyses of univariate estimation-of-distribution algorithms},
  doi       = {10.25932/publishup-43487},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-434870},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xii, 243},
  year      = {2019},
  abstract  = {Optimization is a core part of technological advancement and is usually heavily aided by computers. However, since many optimization problems are hard, it is unrealistic to expect an optimal solution within reasonable time. Hence, heuristics are employed, that is, computer programs that try to produce solutions of high quality quickly. One special class are estimation-of-distribution algorithms (EDAs), which are characterized by maintaining a probabilistic model over the problem domain, which they evolve over time. In an iterative fashion, an EDA uses its model in order to generate a set of solutions, which it then uses to refine the model such that the probability of producing good solutions is increased. In this thesis, we theoretically analyze the class of univariate EDAs over the Boolean domain, that is, over the space of all length-n bit strings. In this setting, the probabilistic model of a univariate EDA consists of an n-dimensional probability vector where each component denotes the probability to sample a 1 for that position in order to generate a bit string. My contribution follows two main directions: first, we analyze general inherent properties of univariate EDAs. Second, we determine the expected run times of specific EDAs on benchmark functions from theory. In the first part, we characterize when EDAs are unbiased with respect to the problem encoding. We then consider a setting where all solutions look equally good to an EDA, and we show that the probabilistic model of an EDA quickly evolves into an incorrect model if it is always updated such that it does not change in expectation. In the second part, we first show that the algorithms cGA and MMAS-fp are able to efficiently optimize a noisy version of the classical benchmark function OneMax. We perturb the function by adding Gaussian noise with a variance of σ², and we prove that the algorithms are able to generate the true optimum in a time polynomial in σ² and the problem size n. For the MMAS-fp, we generalize this result to linear functions. Further, we prove a run time of Ω(n log(n)) for the algorithm UMDA on (unnoisy) OneMax. Last, we introduce a new algorithm that is able to optimize the benchmark functions OneMax and LeadingOnes both in O(n log(n)), which is a novelty for heuristics in the domain we consider.},
  language  = {en}
}
@article{HlawenkaSiemensmeyerWeschkeetal.2018,
  author    = {Hlawenka, Peter and Siemensmeyer, Konrad and Weschke, Eugen and Varykhalov, Andrei and Sanchez-Barriga, Jaime and Shitsevalova, Natalya Y. and Dukhnenko, A. V. and Filipov, V. B. and Gabani, Slavomir and Flachbart, Karol and Rader, Oliver and Rienks, Emile D. L.},
  title     = {Samarium hexaboride is a trivial surface conductor},
  series = {Nature Communications},
  volume    = {9},
  journal   = {Nature Communications},
  publisher = {Nature Publ. Group},
  address   = {London},
  issn      = {2041-1723},
  doi       = {10.1038/s41467-018-02908-7},
  pages     = {7},
  year      = {2018},
  abstract  = {SmB6 is predicted to be the first member of the intersection of topological insulators and Kondo insulators, strongly correlated materials in which the Fermi level lies in the gap of a many-body resonance that forms by hybridization between localized and itinerant states. While robust, surface-only conductivity at low temperature and the observation of surface states at the expected high symmetry points appear to confirm this prediction, we find both surface states at the (100) surface to be topologically trivial. We find the (Gamma) over bar state to appear Rashba split and explain the prominent (X) over bar state by a surface shift of the many-body resonance. We propose that the latter mechanism, which applies to several crystal terminations, can explain the unusual surface conductivity. While additional, as yet unobserved topological surface states cannot be excluded, our results show that a firm connection between the two material classes is still outstanding.},
  language  = {en}
}
@article{PornsawadSungcharoenBoeckmann2020,
  author    = {Pornsawad, Pornsarp and Sungcharoen, Parada and B{\"o}ckmann, Christine},
  title     = {Convergence rate of the modified Landweber method for solving inverse potential problems},
  series = {Mathematics : open access journal},
  volume    = {8},
  journal   = {Mathematics : open access journal},
  number    = {4},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {2227-7390},
  doi       = {10.3390/math8040608},
  pages     = {22},
  year      = {2020},
  abstract  = {In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.},
  language  = {en}
}
@article{GuggenbergerChechkinMetzler2022,
  author    = {Guggenberger, Tobias and Chechkin, Aleksei and Metzler, Ralf},
  title     = {Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {24},
  journal   = {New journal of physics : the open-access journal for physics},
  number    = {7},
  publisher = {Dt. Physikalische Ges.},
  address   = {[Bad Honnef]},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ac7b3c},
  pages     = {18},
  year      = {2022},
  abstract  = {We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.},
  language  = {en}
}
@article{BellingeriFrizPaychaetal.2022,
  author    = {Bellingeri, Carlo and Friz, Peter and Paycha, Sylvie and Preiß, Rosa Lili Dora},
  title     = {Smooth rough paths, their geometry and algebraic renormalization},
  series = {Vietnam journal of mathematics},
  volume    = {50},
  journal   = {Vietnam journal of mathematics},
  number    = {3},
  publisher = {Springer},
  address   = {Singapore},
  issn      = {2305-221X},
  doi       = {10.1007/s10013-022-00570-7},
  pages     = {719 -- 761},
  year      = {2022},
  abstract  = {We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.},
  language  = {en}
}
@article{PetreskaPejovSandevetal.2022,
  author    = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf},
  title     = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting},
  series = {Fractal and fractional},
  volume    = {6},
  journal   = {Fractal and fractional},
  number    = {2},
  publisher = {MDPI AG, Fractal Fract Editorial Office},
  address   = {Basel},
  issn      = {2504-3110},
  doi       = {10.3390/fractalfract6020088},
  pages     = {23},
  year      = {2022},
  abstract  = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.},
  language  = {en}
}
@article{PathirajaMoradkhaniMarshalletal.2018,
  author    = {Pathiraja, Sahani Darschika and Moradkhani, H. and Marshall, L. and Sharma, Ashish and Geenens, G.},
  title     = {Data-driven model uncertainty estimation in hydrologic data assimilation},
  series = {Water resources research : WRR / American Geophysical Union},
  volume    = {54},
  journal   = {Water resources research : WRR / American Geophysical Union},
  number    = {2},
  publisher = {American Geophysical Union},
  address   = {Washington},
  issn      = {0043-1397},
  doi       = {10.1002/2018WR022627},
  pages     = {1252 -- 1280},
  year      = {2018},
  abstract  = {The increasing availability of earth observations necessitates mathematical methods to optimally combine such data with hydrologic models. Several algorithms exist for such purposes, under the umbrella of data assimilation (DA). However, DA methods are often applied in a suboptimal fashion for complex real-world problems, due largely to several practical implementation issues. One such issue is error characterization, which is known to be critical for a successful assimilation. Mischaracterized errors lead to suboptimal forecasts, and in the worst case, to degraded estimates even compared to the no assimilation case. Model uncertainty characterization has received little attention relative to other aspects of DA science. Traditional methods rely on subjective, ad hoc tuning factors or parametric distribution assumptions that may not always be applicable. We propose a novel data-driven approach (named SDMU) to model uncertainty characterization for DA studies where (1) the system states are partially observed and (2) minimal prior knowledge of the model error processes is available, except that the errors display state dependence. It includes an approach for estimating the uncertainty in hidden model states, with the end goal of improving predictions of observed variables. The SDMU is therefore suited to DA studies where the observed variables are of primary interest. Its efficacy is demonstrated through a synthetic case study with low-dimensional chaotic dynamics and a real hydrologic experiment for one-day-ahead streamflow forecasting. In both experiments, the proposed method leads to substantial improvements in the hidden states and observed system outputs over a standard method involving perturbation with Gaussian noise.},
  language  = {en}
}
@article{deWiljesPathirajaReich2020,
  author    = {de Wiljes, Jana and Pathiraja, Sahani Darschika and Reich, Sebastian},
  title     = {Ensemble transform algorithms for nonlinear smoothing problems},
  series = {SIAM journal on scientific computing},
  volume    = {42},
  journal   = {SIAM journal on scientific computing},
  number    = {1},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1064-8275},
  doi       = {10.1137/19M1239544},
  pages     = {A87 -- A114},
  year      = {2020},
  abstract  = {Several numerical tools designed to overcome the challenges of smoothing in a non-linear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter, and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.},
  language  = {en}
}
@article{PathirajaReichStannat2021,
  author    = {Pathiraja, Sahani Darschika and Reich, Sebastian and Stannat, Wilhelm},
  title     = {McKean-Vlasov SDEs in nonlinear filtering},
  series = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics},
  volume    = {59},
  journal   = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics},
  number    = {6},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {0363-0129},
  doi       = {10.1137/20M1355197},
  pages     = {4188 -- 4215},
  year      = {2021},
  abstract  = {Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.},
  language  = {en}
}
@article{ThapaParkKimetal.2022,
  author    = {Thapa, Samudrajit and Park, Seongyu and Kim, Yeongjin and Jeon, Jae-Hyung and Metzler, Ralf and Lomholt, Michael A.},
  title     = {Bayesian inference of scaled versus fractional Brownian motion},
  series = {Journal of physics : A, mathematical and theoretical},
  volume    = {55},
  journal   = {Journal of physics : A, mathematical and theoretical},
  number    = {19},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/ac60e7},
  pages     = {21},
  year      = {2022},
  abstract  = {We present a Bayesian inference scheme for scaled Brownian motion, and investigate its performance on synthetic data for parameter estimation and model selection in a combined inference with fractional Brownian motion. We include the possibility of measurement noise in both models. We find that for trajectories of a few hundred time points the procedure is able to resolve well the true model and parameters. Using the prior of the synthetic data generation process also for the inference, the approach is optimal based on decision theory. We include a comparison with inference using a prior different from the data generating one.},
  language  = {en}
}
@article{PathirajaAnghileriBurlandoetal.2018,
  author    = {Pathiraja, Sahani Darschika and Anghileri, Daniela and Burlando, Paolo and Sharma, Ashish and Marshall, Lucy and Moradkhani, Hamid},
  title     = {Time-varying parameter models for catchments with land use change},
  series = {Hydrology and earth system sciences : HESS},
  volume    = {22},
  journal   = {Hydrology and earth system sciences : HESS},
  number    = {5},
  publisher = {Copernicus},
  address   = {G{\"o}ttingen},
  issn      = {1027-5606},
  doi       = {10.5194/hess-22-2903-2018},
  pages     = {2903 -- 2919},
  year      = {2018},
  abstract  = {Rapid population and economic growth in Southeast Asia has been accompanied by extensive land use change with consequent impacts on catchment hydrology. Modeling methodologies capable of handling changing land use conditions are therefore becoming ever more important and are receiving increasing attention from hydrologists. A recently developed data-assimilation-based framework that allows model parameters to vary through time in response to signals of change in observations is considered for a medium-sized catchment (2880 km(2)) in northern Vietnam experiencing substantial but gradual land cover change. We investigate the efficacy of the method as well as the importance of the chosen model structure in ensuring the success of a time-varying parameter method. The method was used with two lumped daily conceptual models (HBV and HyMOD) that gave good-quality streamflow predictions during pre-change conditions. Although both time-varying parameter models gave improved streamflow predictions under changed conditions compared to the time-invariant parameter model, persistent biases for low flows were apparent in the HyMOD case. It was found that HyMOD was not suited to representing the modified baseflow conditions, resulting in extreme and unrealistic time-varying parameter estimates. This work shows that the chosen model can be critical for ensuring the time-varying parameter framework successfully models streamflow under changing land cover conditions. It can also be used to determine whether land cover changes (and not just meteorological factors) contribute to the observed hydrologic changes in retrospective studies where the lack of a paired control catchment precludes such an assessment.},
  language  = {en}
}
@article{PathirajaAnghileriBurlandoetal.2017,
  author    = {Pathiraja, Sahani Darschika and Anghileri, Daniela and Burlando, P. and Sharma, A. and Marshall, L. and Moradkhani, H.},
  title     = {Insights on the impact of systematic model errors on data assimilation performance in changing catchments},
  series = {Advances in water resources},
  volume    = {113},
  journal   = {Advances in water resources},
  publisher = {Elsevier},
  address   = {Oxford},
  issn      = {0309-1708},
  doi       = {10.1016/j.advwatres.2017.12.006},
  pages     = {202 -- 222},
  year      = {2017},
  abstract  = {The global prevalence of rapid and extensive land use change necessitates hydrologic modelling methodologies capable of handling non-stationarity. This is particularly true in the context of Hydrologic Forecasting using Data Assimilation. Data Assimilation has been shown to dramatically improve forecast skill in hydrologic and meteorological applications, although such improvements are conditional on using bias-free observations and model simulations. A hydrologic model calibrated to a particular set of land cover conditions has the potential to produce biased simulations when the catchment is disturbed. This paper sheds new light on the impacts of bias or systematic errors in hydrologic data assimilation, in the context of forecasting in catchments with changing land surface conditions and a model calibrated to pre-change conditions. We posit that in such cases, the impact of systematic model errors on assimilation or forecast quality is dependent on the inherent prediction uncertainty that persists even in pre-change conditions. Through experiments on a range of catchments, we develop a conceptual relationship between total prediction uncertainty and the impacts of land cover changes on the hydrologic regime to demonstrate how forecast quality is affected when using state estimation Data Assimilation with no modifications to account for land cover changes. This work shows that systematic model errors as a result of changing or changed catchment conditions do not always necessitate adjustments to the modelling or assimilation methodology, for instance through re-calibration of the hydrologic model, time varying model parameters or revised offline/online bias estimation.},
  language  = {en}
}
@misc{KolbeEvans2020,
  author    = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.},
  title     = {Isotopic tiling theory for hyperbolic surfaces},
  series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-54428},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-544285},
  pages     = {30},
  year      = {2020},
  abstract  = {In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.},
  language  = {en}
}
@article{LeungLeutbecherReichetal.2021,
  author    = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.},
  title     = {Forecast verification},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {147},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {739},
  publisher = {Wiley},
  address   = {Hoboken},
  issn      = {0035-9009},
  doi       = {10.1002/qj.4120},
  pages     = {3124 -- 3134},
  year      = {2021},
  abstract  = {The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS-RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met.},
  language  = {en}
}
@article{AyanbayevKlebanovLietal.2021,
  author    = {Ayanbayev, Birzhan and Klebanov, Ilja and Li, Han Cheng and Sullivan, Tim J.},
  title     = {Gamma-convergence of Onsager-Machlup functionals},
  series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  volume    = {38},
  journal   = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0266-5611},
  doi       = {10.1088/1361-6420/ac3f81},
  pages     = {32},
  year      = {2021},
  abstract  = {The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.},
  language  = {en}
}
@article{LieStahnSullivan2022,
  author    = {Lie, Han Cheng and Stahn, Martin and Sullivan, Tim J.},
  title     = {Randomised one-step time integration methods for deterministic operator differential equations},
  series = {Calcolo},
  volume    = {59},
  journal   = {Calcolo},
  number    = {1},
  publisher = {Springer},
  address   = {Milano},
  issn      = {0008-0624},
  doi       = {10.1007/s10092-022-00457-6},
  pages     = {33},
  year      = {2022},
  abstract  = {Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.},
  language  = {en}
}
@article{MalemShinitskiOjedaOpper2022,
  author    = {Malem-Shinitski, Noa and Ojeda, Cesar and Opper, Manfred},
  title     = {Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects},
  series = {Entropy},
  volume    = {24},
  journal   = {Entropy},
  number    = {3},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e24030356},
  pages     = {22},
  year      = {2022},
  abstract  = {Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results.},
  language  = {en}
}
@phdthesis{MalemShinitski2023,
  author    = {Malem-Shinitski, Noa},
  title     = {Bayesian inference and modeling for point processes with applications from neuronal activity to scene viewing},
  doi       = {10.25932/publishup-61495},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-614952},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 129},
  year      = {2023},
  abstract  = {Point processes are a common methodology to model sets of events. From earthquakes to social media posts, from the arrival times of neuronal spikes to the timing of crimes, from stock prices to disease spreading -- these phenomena can be reduced to the occurrences of events concentrated in points. Often, these events happen one after the other defining a time--series. Models of point processes can be used to deepen our understanding of such events and for classification and prediction. Such models include an underlying random process that generates the events. This work uses Bayesian methodology to infer the underlying generative process from observed data. Our contribution is twofold -- we develop new models and new inference methods for these processes. We propose a model that extends the family of point processes where the occurrence of an event depends on the previous events. This family is known as Hawkes processes. Whereas in most existing models of such processes, past events are assumed to have only an excitatory effect on future events, we focus on the newly developed nonlinear Hawkes process, where past events could have excitatory and inhibitory effects. After defining the model, we present its inference method and apply it to data from different fields, among others, to neuronal activity. The second model described in the thesis concerns a specific instance of point processes --- the decision process underlying human gaze control. This process results in a series of fixated locations in an image. We developed a new model to describe this process, motivated by the known Exploration--Exploitation dilemma. Alongside the model, we present a Bayesian inference algorithm to infer the model parameters. Remaining in the realm of human scene viewing, we identify the lack of best practices for Bayesian inference in this field. We survey four popular algorithms and compare their performances for parameter inference in two scan path models. The novel models and inference algorithms presented in this dissertation enrich the understanding of point process data and allow us to uncover meaningful insights.},
  language  = {en}
}
@article{AyanbayevKlebanovLieetal.2021,
  author    = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, Tim J.},
  title     = {Gamma-convergence of Onsager-Machlup functionals},
  series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  volume    = {38},
  journal   = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0266-5611},
  doi       = {10.1088/1361-6420/ac3f82},
  pages     = {35},
  year      = {2021},
  abstract  = {We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.},
  language  = {en}
}
@misc{EhlenFloegeGoebeletal.2023,
  author    = {Ehlen, Tobias and Fl{\"o}ge, Annie and G{\"o}bel, Franziska and Keller, Peter and Rœlly, Sylvie},
  title     = {{\"U}bungsbuch zur Stochastik},
  editor    = {Keller, Peter and Rœlly, Sylvie},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-563-7},
  doi       = {10.25932/publishup-59593},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-595939},
  pages     = {306},
  year      = {2023},
  abstract  = {Dieses Buch stellt {\"U}bungen zu den Grundbegriffen und Grunds{\"a}tzen der Stochastik und ihre L{\"o}sungen zur Verf{\"u}gung. So wie man Tonleitern in der Musik trainiert, so berechnet man {\"U}bungsaufgaben in der Mathematik. In diesem Sinne soll dieses {\"U}bungsbuch vor allem als Vorlage dienen f{\"u}r das eigenst{\"a}ndige, eigenverantwortliche Lernen und {\"U}ben. Die Sch{\"o}nheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Ph{\"a}nomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Gl{\"u}cksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.},
  language  = {de}
}
@article{LazaridesRubach2017,
  author    = {Lazarides, Rebecca and Rubach, Charlott},
  title     = {Instructional characteristics in mathematics classrooms},
  series = {Mathematics Education Research Journal},
  volume    = {29},
  journal   = {Mathematics Education Research Journal},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {1033-2170},
  doi       = {10.1007/s13394-017-0196-4},
  pages     = {201 -- 217},
  year      = {2017},
  abstract  = {This longitudinal study examined relationships between student-perceived teaching for meaning, support for autonomy, and competence in mathematic classrooms (Time 1), and students' achievement goal orientations and engagement in mathematics 6 months later (Time 2). We tested whether student-perceived instructional characteristics at Time 1 indirectly related to student engagement at Time 2, via their achievement goal orientations (Time 2), and, whether student gender moderated these relationships. Participants were ninth and tenth graders (55.2\% girls) from 46 classrooms in ten secondary schools in Berlin, Germany. Only data from students who participated at both timepoints were included (N = 746 out of total at Time 1 1118; dropout 33.27\%). Longitudinal structural equation modeling showed that student-perceived teaching for meaning and support for competence indirectly predicted intrinsic motivation and effort, via students' mastery goal orientation. These paths were equivalent for girls and boys. The findings are significant for mathematics education, in identifying motivational processes that partly explain the relationships between student-perceived teaching for meaning and competence support and intrinsic motivation and effort in mathematics.},
  language  = {en}
}
@article{EshghiMachReichel2021,
  author    = {Eshghi, Nasim and Mach, Thomas and Reichel, Lothar},
  title     = {New matrix function approximations and quadrature rules based on the Arnoldi process},
  series = {Journal of computational and applied mathematics},
  volume    = {391},
  journal   = {Journal of computational and applied mathematics},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0377-0427},
  doi       = {10.1016/j.cam.2021.113442},
  pages     = {12},
  year      = {2021},
  abstract  = {The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f (A)v and matrix functionals of the form v*(f (A))*g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript * denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v*(f (A))*g(A)v about as much as performing an additional step of the Arnoldi process.},
  language  = {en}
}
@book{ZassZagrebnovSukiasyanetal.2020,
  author    = {Zass, Alexander and Zagrebnov, Valentin and Sukiasyan, Hayk and Melkonyan, Tatev and Rafler, Mathias and Poghosyan, Suren and Zessin, Hans and Piatnitski, Andrey and Zhizhina, Elena and Pechersky, Eugeny and Pirogov, Sergei and Yambartsev, Anatoly and Mazzonetto, Sara and Lykov, Alexander and Malyshev, Vadim and Khachatryan, Linda and Nahapetian, Boris and Jursenas, Rytis and Jansen, Sabine and Tsagkarogiannis, Dimitrios and Kuna, Tobias and Kolesnikov, Leonid and Hryniv, Ostap and Wallace, Clare and Houdebert, Pierre and Figari, Rodolfo and Teta, Alessandro and Boldrighini, Carlo and Frigio, Sandro and Maponi, Pierluigi and Pellegrinotti, Alessandro and Sinai, Yakov G.},
  title     = {Proceedings of the XI international conference stochastic and analytic methods in mathematical physics},
  number    = {6},
  editor    = {Roelly, Sylvie and Rafler, Mathias and Poghosyan, Suren},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-485-2},
  issn      = {2199-4951},
  doi       = {10.25932/publishup-45919},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-459192},
  publisher      = {Universit{\"a}t Potsdam},
  pages     = {xiv, 194},
  year      = {2020},
  abstract  = {The XI international conference Stochastic and Analytic Methods in Mathematical Physics was held in Yerevan 2 - 7 September 2019 and was dedicated to the memory of the great mathematician Robert Adol'fovich Minlos, who passed away in January 2018. The present volume collects a large majority of the contributions presented at the conference on the following domains of contemporary interest: classical and quantum statistical physics, mathematical methods in quantum mechanics, stochastic analysis, applications of point processes in statistical mechanics. The authors are specialists from Armenia, Czech Republic, Denmark, France, Germany, Italy, Japan, Lithuania, Russia, UK and Uzbekistan. A particular aim of this volume is to offer young scientists basic material in order to inspire their future research in the wide fields presented here.},
  language  = {en}
}
@article{Omel'chenkoLaing2022,
  author    = {Omel'chenko, Oleh and Laing, Carlo R.},
  title     = {Collective states in a ring network of theta neurons},
  series = {Proceedings of the Royal Society of London. Series A, Mathematical, physical and engineering sciences},
  volume    = {478},
  journal   = {Proceedings of the Royal Society of London. Series A, Mathematical, physical and engineering sciences},
  number    = {2259},
  publisher = {Royal Society},
  address   = {London},
  issn      = {1364-5021},
  doi       = {10.1098/rspa.2021.0817},
  pages     = {23},
  year      = {2022},
  abstract  = {We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results.},
  language  = {en}
}
@article{KolbeEvans2022,
  author    = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.},
  title     = {Enumerating isotopy classes of tilings guided by the symmetry of triply},
  series = {Siam journal on applied algebra and geometry},
  volume    = {6},
  journal   = {Siam journal on applied algebra and geometry},
  number    = {1},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2470-6566},
  doi       = {10.1137/20M1358943},
  pages     = {1 -- 40},
  year      = {2022},
  abstract  = {We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.},
  language  = {en}
}
@phdthesis{Wallenta2015,
  author    = {Wallenta, Daniel},
  title     = {Sequences of compact curvature},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-87489},
  school      = {Universit{\"a}t Potsdam},
  pages     = {viii, 73},
  year      = {2015},
  abstract  = {By perturbing the differential of a (cochain-)complex by "small" operators, one obtains what is referred to as quasicomplexes, i.e. a sequence whose curvature is not equal to zero in general. In this situation the cohomology is no longer defined. Note that it depends on the structure of the underlying spaces whether or not an operator is "small." This leads to a magical mix of perturbation and regularisation theory. In the general setting of Hilbert spaces compact operators are "small." In order to develop this theory, many elements of diverse mathematical disciplines, such as functional analysis, differential geometry, partial differential equation, homological algebra and topology have to be combined. All essential basics are summarised in the first chapter of this thesis. This contains classical elements of index theory, such as Fredholm operators, elliptic pseudodifferential operators and characteristic classes. Moreover we study the de Rham complex and introduce Sobolev spaces of arbitrary order as well as the concept of operator ideals. In the second chapter, the abstract theory of (Fredholm) quasicomplexes of Hilbert spaces will be developed. From the very beginning we will consider quasicomplexes with curvature in an ideal class. We introduce the Euler characteristic, the cone of a quasiendomorphism and the Lefschetz number. In particular, we generalise Euler's identity, which will allow us to develop the Lefschetz theory on nonseparable Hilbert spaces. Finally, in the third chapter the abstract theory will be applied to elliptic quasicomplexes with pseudodifferential operators of arbitrary order. We will show that the Atiyah-Singer index formula holds true for those objects and, as an example, we will compute the Euler characteristic of the connection quasicomplex. In addition to this we introduce geometric quasiendomorphisms and prove a generalisation of the Lefschetz fixed point theorem of Atiyah and Bott.},
  language  = {en}
}
@phdthesis{Wichitsanguan2016,
  author    = {Wichitsa-nguan, Korakot},
  title     = {Modifications and extensions of the logistic regression and Cox model},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-90033},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 131},
  year      = {2016},
  abstract  = {In many statistical applications, the aim is to model the relationship between covariates and some outcomes. A choice of the appropriate model depends on the outcome and the research objectives, such as linear models for continuous outcomes, logistic models for binary outcomes and the Cox model for time-to-event data. In epidemiological, medical, biological, societal and economic studies, the logistic regression is widely used to describe the relationship between a response variable as binary outcome and explanatory variables as a set of covariates. However, epidemiologic cohort studies are quite expensive regarding data management since following up a large number of individuals takes long time. Therefore, the case-cohort design is applied to reduce cost and time for data collection. The case-cohort sampling collects a small random sample from the entire cohort, which is called subcohort. The advantage of this design is that the covariate and follow-up data are recorded only on the subcohort and all cases (all members of the cohort who develop the event of interest during the follow-up process). In this thesis, we investigate the estimation in the logistic model for case-cohort design. First, a model with a binary response and a binary covariate is considered. The maximum likelihood estimator (MLE) is described and its asymptotic properties are established. An estimator for the asymptotic variance of the estimator based on the maximum likelihood approach is proposed; this estimator differs slightly from the estimator introduced by Prentice (1986). Simulation results for several proportions of the subcohort show that the proposed estimator gives lower empirical bias and empirical variance than Prentice's estimator. Then the MLE in the logistic regression with discrete covariate under case-cohort design is studied. Here the approach of the binary covariate model is extended. Proving asymptotic normality of estimators, standard errors for the estimators can be derived. The simulation study demonstrates the estimation procedure of the logistic regression model with a one-dimensional discrete covariate. Simulation results for several proportions of the subcohort and different choices of the underlying parameters indicate that the estimator developed here performs reasonably well. Moreover, the comparison between theoretical values and simulation results of the asymptotic variance of estimator is presented. Clearly, the logistic regression is sufficient for the binary outcome refers to be available for all subjects and for a fixed time interval. Nevertheless, in practice, the observations in clinical trials are frequently collected for different time periods and subjects may drop out or relapse from other causes during follow-up. Hence, the logistic regression is not appropriate for incomplete follow-up data; for example, an individual drops out of the study before the end of data collection or an individual has not occurred the event of interest for the duration of the study. These observations are called censored observations. The survival analysis is necessary to solve these problems. Moreover, the time to the occurence of the event of interest is taken into account. The Cox model has been widely used in survival analysis, which can effectively handle the censored data. Cox (1972) proposed the model which is focused on the hazard function. The Cox model is assumed to be λ(t|x) = λ0(t) exp(β^Tx) where λ0(t) is an unspecified baseline hazard at time t and X is the vector of covariates, β is a p-dimensional vector of coefficient. In this thesis, the Cox model is considered under the view point of experimental design. The estimability of the parameter β0 in the Cox model, where β0 denotes the true value of β, and the choice of optimal covariates are investigated. We give new representations of the observed information matrix In(β) and extend results for the Cox model of Andersen and Gill (1982). In this way conditions for the estimability of β0 are formulated. Under some regularity conditions, ∑ is the inverse of the asymptotic variance matrix of the MPLE of β0 in the Cox model and then some properties of the asymptotic variance matrix of the MPLE are highlighted. Based on the results of asymptotic estimability, the calculation of local optimal covariates is considered and shown in examples. In a sensitivity analysis, the efficiency of given covariates is calculated. For neighborhoods of the exponential models, the efficiencies have then been found. It is appeared that for fixed parameters β0, the efficiencies do not change very much for different baseline hazard functions. Some proposals for applicable optimal covariates and a calculation procedure for finding optimal covariates are discussed. Furthermore, the extension of the Cox model where time-dependent coefficient are allowed, is investigated. In this situation, the maximum local partial likelihood estimator for estimating the coefficient function β(·) is described. Based on this estimator, we formulate a new test procedure for testing, whether a one-dimensional coefficient function β(·) has a prespecified parametric form, say β(·; ϑ). The score function derived from the local constant partial likelihood function at d distinct grid points is considered. It is shown that the distribution of the properly standardized quadratic form of this d-dimensional vector under the null hypothesis tends to a Chi-squared distribution. Moreover, the limit statement remains true when replacing the unknown ϑ0 by the MPLE in the hypothetical model and an asymptotic α-test is given by the quantiles or p-values of the limiting Chi-squared distribution. Finally, we propose a bootstrap version of this test. The bootstrap test is only defined for the special case of testing whether the coefficient function is constant. A simulation study illustrates the behavior of the bootstrap test under the null hypothesis and a special alternative. It gives quite good results for the chosen underlying model. References P. K. Andersen and R. D. Gill. Cox's regression model for counting processes: a large samplestudy. Ann. Statist., 10(4):1100{1120, 1982. D. R. Cox. Regression models and life-tables. J. Roy. Statist. Soc. Ser. B, 34:187{220, 1972. R. L. Prentice. A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika, 73(1):1{11, 1986.},
  language  = {en}
}
@unpublished{MakhmudovTarkhanov2014,
  author    = {Makhmudov, Olimdjan and Tarkhanov, Nikolai Nikolaevich},
  title     = {The first mixed problem for the nonstationary Lam{\´e} system},
  volume    = {3},
  number    = {10},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71923},
  pages     = {19},
  year      = {2014},
  abstract  = {We find an adequate interpretation of the Lam{\´e} operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lam{\´e} system.},
  language  = {en}
}
@phdthesis{Ziese2014,
  author    = {Ziese, Ramona},
  title     = {Geometric electroelasticity},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72504},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vi, 113},
  year      = {2014},
  abstract  = {In this work a diffential geometric formulation of the theory of electroelasticity is developed which also includes thermal and magnetic influences. We study the motion of bodies consisting of an elastic material that are deformed by the influence of mechanical forces, heat and an external electromagnetic field. To this end physical balance laws (conservation of mass, balance of momentum, angular momentum and energy) are established. These provide an equation that describes the motion of the body during the deformation. Here the body and the surrounding space are modeled as Riemannian manifolds, and we allow that the body has a lower dimension than the surrounding space. In this way one is not (as usual) restricted to the description of the deformation of three-dimensional bodies in a three-dimensional space, but one can also describe the deformation of membranes and the deformation in a curved space. Moreover, we formulate so-called constitutive relations that encode the properties of the used material. Balance of energy as a scalar law can easily be formulated on a Riemannian manifold. The remaining balance laws are then obtained by demanding that balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space. This generalizes a result by Marsden and Hughes that pertains to bodies that have the same dimension as the surrounding space and does not allow the presence of electromagnetic fields. Usually, in works on electroelasticity the entropy inequality is used to decide which otherwise allowed deformations are physically admissible and which are not. It is alsoemployed to derive restrictions to the possible forms of constitutive relations describing the material. Unfortunately, the opinions on the physically correct statement of the entropy inequality diverge when electromagnetic fields are present. Moreover, it is unclear how to formulate the entropy inequality in the case of a membrane that is subjected to an electromagnetic field. Thus, we show that one can replace the use of the entropy inequality by the demand that for a given process balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space and under linear rescalings of the temperature. On the one hand, this demand also yields the desired restrictions to the form of the constitutive relations. On the other hand, it needs much weaker assumptions than the arguments in physics literature that are employing the entropy inequality. Again, our result generalizes a theorem of Marsden and Hughes. This time, our result is, like theirs, only valid for bodies that have the same dimension as the surrounding space.},
  language  = {en}
}
@phdthesis{Vu2014,
  author    = {Vu, Dinh Phuong},
  title     = {Using video study to investigate eighth-grade mathematics classrooms in Vietnam},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72464},
  school      = {Universit{\"a}t Potsdam},
  pages     = {273},
  year      = {2014},
  abstract  = {The International Project for the Evaluation of Educational Achievement (IEA) was formed in the 1950s (Postlethwaite, 1967). Since that time, the IEA has conducted many studies in the area of mathematics, such as the First International Mathematics Study (FIMS) in 1964, the Second International Mathematics Study (SIMS) in 1980-1982, and a series of studies beginning with the Third International Mathematics and Science Study (TIMSS) which has been conducted every 4 years since 1995. According to Stigler et al. (1999), in the FIMS and the SIMS, U.S. students achieved low scores in comparison with students in other countries (p. 1). The TIMSS 1995 "Videotape Classroom Study" was therefore a complement to the earlier studies conducted to learn "more about the instructional and cultural processes that are associated with achievement" (Stigler et al., 1999, p. 1). The TIMSS Videotape Classroom Study is known today as the TIMSS Video Study. From the findings of the TIMSS 1995 Video Study, Stigler and Hiebert (1999) likened teaching to "mountain ranges poking above the surface of the water," whereby they implied that we might see the mountaintops, but we do not see the hidden parts underneath these mountain ranges (pp. 73-78). By watching the videotaped lessons from Germany, Japan, and the United States again and again, they discovered that "the systems of teaching within each country look similar from lesson to lesson. At least, there are certain recurring features [or patterns] that typify many of the lessons within a country and distinguish the lessons among countries" (pp. 77-78). They also discovered that "teaching is a cultural activity," so the systems of teaching "must be understood in relation to the cultural beliefs and assumptions that surround them" (pp. 85, 88). From this viewpoint, one of the purposes of this dissertation was to study some cultural aspects of mathematics teaching and relate the results to mathematics teaching and learning in Vietnam. Another research purpose was to carry out a video study in Vietnam to find out the characteristics of Vietnamese mathematics teaching and compare these characteristics with those of other countries. In particular, this dissertation carried out the following research tasks: - Studying the characteristics of teaching and learning in different cultures and relating the results to mathematics teaching and learning in Vietnam - Introducing the TIMSS, the TIMSS Video Study and the advantages of using video study in investigating mathematics teaching and learning - Carrying out the video study in Vietnam to identify the image, scripts and patterns, and the lesson signature of eighth-grade mathematics teaching in Vietnam - Comparing some aspects of mathematics teaching in Vietnam and other countries and identifying the similarities and differences across countries - Studying the demands and challenges of innovating mathematics teaching methods in Vietnam - lessons from the video studies Hopefully, this dissertation will be a useful reference material for pre-service teachers at education universities to understand the nature of teaching and develop their teaching career.},
  language  = {en}
}
@unpublished{BagdonavičiusLevulieneNikulinetal.2004,
  author    = {Bagdonavičius, Vilijandas B. and Levuliene, Ruta and Nikulin, Mikhail S. and Zdorova-Cheminade, Olga},
  title     = {Tests for homogeneity of survival distributions against non-location alternatives and analysis of the gastric cancer data},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51527},
  year      = {2004},
  abstract  = {The two and k-sample tests of equality of the survival distributions against the alternatives including cross-effects of survival functions, proportional and monotone hazard ratios, are given for the right censored data. The asymptotic power against approaching alternatives is investigated. The tests are applied to the well known chemio and radio therapy data of the Gastrointestinal Tumor Study Group. The P-values for both proposed tests are much smaller then in the case of other known tests. Differently from the test of Stablein and Koutrouvelis the new tests can be applied not only for singly but also to randomly censored data.},
  language  = {en}
}
@unpublished{GairingHoegeleKosenkovaetal.2014,
  author    = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana and Kulik, Alexei Michajlovič},
  title     = {On the calibration of L{\´e}vy driven time series with coupling distances : an application in paleoclimate},
  volume    = {3},
  number    = {2},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69781},
  pages     = {18},
  year      = {2014},
  abstract  = {This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a L{\´e}vy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noisy behavior of the data and a given reference jump measure in terms of so-called coupling distances. After a short introduction to L{\´e}vy processes and coupling distances we recall basic statistical approximation results and derive rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study to simulated and paleoclimate data. It indicates the dominant presence of a non-stable heavy-tailed jump L{\´e}vy component for some tail index greater than 2.},
  language  = {en}
}
@unpublished{AizenbergTarkhanov2014,
  author    = {Aizenberg, Lev A. and Tarkhanov, Nikolai Nikolaevich},
  title     = {An integral formula for the number of lattice points in a domain},
  volume    = {3},
  number    = {3},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70453},
  pages     = {7},
  year      = {2014},
  abstract  = {Using the multidimensional logarithmic residue we show a simple formula for the difference between the number of integer points in a bounded domain of R^n and the volume of this domain. The difference proves to be the integral of an explicit differential form over the boundary of the domain.},
  language  = {en}
}
@unpublished{Wallenta2013,
  author    = {Wallenta, Daniel},
  title     = {A Lefschetz fixed point formula for elliptic quasicomplexes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67016},
  year      = {2013},
  abstract  = {In a recent paper with N. Tarkhanov, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.},
  language  = {en}
}
@unpublished{HoegelePavlyukevich2014,
  author    = {H{\"o}gele, Michael and Pavlyukevich, Ilya},
  title     = {Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed L{\´e}vy type noise},
  volume    = {3},
  number    = {5},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70639},
  pages     = {27},
  year      = {2014},
  abstract  = {We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse-Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed L{\´e}vy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a It{\^o}, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.},
  language  = {en}
}
@unpublished{ConfortiDaiPraRoelly2014,
  author    = {Conforti, Giovanni and Dai Pra, Paolo and Roelly, Sylvie},
  title     = {Reciprocal class of jump processes},
  volume    = {3},
  number    = {6},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70776},
  pages     = {30},
  year      = {2014},
  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 A in R^d. 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 A 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}
}
@misc{Huebner2021,
  type      = {Master Thesis},
  author    = {H{\"u}bner, Andrea},
  title     = {Ein multityper Verzweigungsprozess als Modell zur Untersuchung der Ausbreitung von Covid-19},
  doi       = {10.25932/publishup-50922},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-509225},
  school      = {Universit{\"a}t Potsdam},
  year      = {2021},
  abstract  = {Im Zuge der Covid-19 Pandemie werden zwei Werte t{\"a}glich diskutiert: Die zuletzt gemeldete Zahl der neu Infizierten und die sogenannte Reproduktionsrate. Sie gibt wieder, wie viele weitere Menschen ein an Corona erkranktes Individuum im Durchschnitt ansteckt. F{\"u}r die Sch{\"a}tzung dieses Wertes gibt es viele M{\"o}glichkeiten - auch das Robert Koch-Institut gibt in seinem t{\"a}glichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere M{\"o}glichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu sch{\"a}tzen. In der ersten H{\"a}lfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man f{\"u}r die Modellierung ben{\"o}tigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverst{\"a}ndnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingef{\"u}hrt und die Ergebnisse aus diesem Themengebiet, die f{\"u}r diese Arbeit wichtig sind, pr{\"a}sentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschr{\"a}nken wir uns auf zwei Typen. Das Prinzip l{\"a}sst sich aber auf eine beliebige Anzahl von Typen erweitern. Vor allem soll die Wichtigkeit des Parameters λ herausgestellt werden. Dieser Wert kann als durchschnittliche Zahl von Nachfahren eines Individuums interpretiert werden und bestimmt die Dynamik des Prozesses {\"u}ber einen l{\"a}ngeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R. In der zweiten H{\"a}lfte dieser Arbeit stellen wir eine Anwendung der Theorie {\"u}ber Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Ver{\"o}ffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' {\"u}ber einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Sch{\"a}tzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die M{\"o}glichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsf{\"a}lle) zu sch{\"a}tzen. Wir wenden die Sch{\"a}tzer auf die Zahlen von Deutschland an und werten diese schließlich aus.},
  language  = {de}
}
@unpublished{DyachenkoTarkhanov2014,
  author    = {Dyachenko, Evgueniya and Tarkhanov, Nikolai Nikolaevich},
  title     = {Singular perturbations of elliptic operators},
  volume    = {3},
  number    = {1},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69502},
  pages     = {21},
  year      = {2014},
  abstract  = {We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.},
  language  = {en}
}
@phdthesis{Samaras2016,
  author    = {Samaras, Stefanos},
  title     = {Microphysical retrieval of non-spherical aerosol particles using regularized inversion of multi-wavelength lidar data},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-396528},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xiv, 190},
  year      = {2016},
  abstract  = {Numerous reports of relatively rapid climate changes over the past century make a clear case of the impact of aerosols and clouds, identified as sources of largest uncertainty in climate projections. Earth's radiation balance is altered by aerosols depending on their size, morphology and chemical composition. Competing effects in the atmosphere can be further studied by investigating the evolution of aerosol microphysical properties, which are the focus of the present work. The aerosol size distribution, the refractive index, and the single scattering albedo are commonly used such properties linked to aerosol type, and radiative forcing. Highly advanced lidars (light detection and ranging) have reduced aerosol monitoring and optical profiling into a routine process. Lidar data have been widely used to retrieve the size distribution through the inversion of the so-called Lorenz-Mie model (LMM). This model offers a reasonable treatment for spherically approximated particles, it no longer provides, though, a viable description for other naturally occurring arbitrarily shaped particles, such as dust particles. On the other hand, non-spherical geometries as simple as spheroids reproduce certain optical properties with enhanced accuracy. Motivated by this, we adapt the LMM to accommodate the spheroid-particle approximation introducing the notion of a two-dimensional (2D) shape-size distribution. Inverting only a few optical data points to retrieve the shape-size distribution is classified as a non-linear ill-posed problem. A brief mathematical analysis is presented which reveals the inherent tendency towards highly oscillatory solutions, explores the available options for a generalized solution through regularization methods and quantifies the ill-posedness. The latter will improve our understanding on the main cause fomenting instability in the produced solution spaces. The new approach facilitates the exploitation of additional lidar data points from depolarization measurements, associated with particle non-sphericity. However, the generalization of LMM vastly increases the complexity of the problem. The underlying theory for the calculation of the involved optical cross sections (T-matrix theory) is computationally so costly, that would limit a retrieval analysis to an unpractical point. Moreover the discretization of the model equation by a 2D collocation method, proposed in this work, involves double integrations which are further time consuming. We overcome these difficulties by using precalculated databases and a sophisticated retrieval software (SphInX: Spheroidal Inversion eXperiments) especially developed for our purposes, capable of performing multiple-dataset inversions and producing a wide range of microphysical retrieval outputs. Hybrid regularization in conjunction with minimization processes is used as a basis for our algorithms. Synthetic data retrievals are performed simulating various atmospheric scenarios in order to test the efficiency of different regularization methods. The gap in contemporary literature in providing full sets of uncertainties in a wide variety of numerical instances is of major concern here. For this, the most appropriate methods are identified through a thorough analysis on an overall-behavior basis regarding accuracy and stability. The general trend of the initial size distributions is captured in our numerical experiments and the reconstruction quality depends on data error level. Moreover, the need for more or less depolarization points is explored for the first time from the point of view of the microphysical retrieval. Finally, our approach is tested in various measurement cases giving further insight for future algorithm improvements.},
  language  = {en}
}
@phdthesis{Santilli2017,
  author    = {Santilli, Mario},
  title     = {Higher order rectifiability in Euclidean space},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-403632},
  school      = {Universit{\"a}t Potsdam},
  pages     = {45},
  year      = {2017},
  abstract  = {The first main goal of this thesis is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for every integer k ≥ 2 we introduce the approximate differential of order k of A and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications. The other goal is to extend to general closed sets a well known theorem of Alberti on the second order rectifiability properties of the boundary of convex bodies. The Alberti theorem provides a stratification of second order rectifiable subsets of the boundary of a convex body based on the dimension of the (convex) normal cone. Considering a suitable generalization of this normal cone for general closed subsets of the Euclidean space and employing some results from the first part we can prove that the same stratification exists for every closed set.},
  language  = {en}
}
@unpublished{BagderinaTarkhanov2013,
  author    = {Bagderina, Yulia Yu. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Differential invariants of a class of Lagrangian systems with two degrees of freedom},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63129},
  year      = {2013},
  abstract  = {We consider systems of Euler-Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie's infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.},
  language  = {en}
}
@unpublished{BlanchardDelattreRoquain2012,
  author    = {Blanchard, Gilles and Delattre, Sylvain and Roquain, {\´E}tienne},
  title     = {Testing over a continuum of null hypotheses},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56877},
  year      = {2012},
  abstract  = {We introduce a theoretical framework for performing statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum. As a measure of type I error, we extend the concept of false discovery rate (FDR) to this setting. The FDR is defined as the average ratio of the measure of two random sets, so that its study presents some challenge and is of some intrinsic mathematical interest. Our main result shows how to use the p-value process to control the FDR at a nominal level, either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting, the latter one leading to a less conservative procedure. The interest of this approach is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. Conceptually, an interesting feature of the setting advocated here is that it focuses directly on the intrinsic hypothesis space associated with a testing model on a random process, without referring to an arbitrary discretization.},
  language  = {en}
}
@unpublished{GrudskyTarkhanov2012,
  author    = {Grudsky, Serguey and Tarkhanov, Nikolai Nikolaevich},
  title     = {Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57745},
  year      = {2012},
  abstract  = {We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.},
  language  = {en}
}
@unpublished{KleinRosenberger2012,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Tunneling for a class of difference operators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56989},
  year      = {2012},
  abstract  = {We analyze a general class of difference operators containing a multi-well potential and a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we treat the eigenvalue problem as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix similar to the analysis for the Schr{\"o}dinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.},
  language  = {en}
}
@unpublished{KellerRoellyValleriani2012,
  author    = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo},
  title     = {On time duality for quasi-birth-and-death processes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56973},
  year      = {2012},
  abstract  = {We say that (weak/strong) time duality holds for continuous time quasi-birth-and-death-processes if, starting from a fixed level, the first hitting time of the next upper level and the first hitting time of the next lower level have the same distribution. We present here a criterion for time duality in the case where transitions from one level to another have to pass through a given single state, the so-called bottleneck property. We also prove that a weaker form of reversibility called balanced under permutation is sufficient for the time duality to hold. We then discuss the general case.},
  language  = {en}
}
@unpublished{TarkhanovWallenta2012,
  author    = {Tarkhanov, Nikolai Nikolaevich and Wallenta, Daniel},
  title     = {The Lefschetz number of sequences of trace class curvature},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56969},
  year      = {2012},
  abstract  = {For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.},
  language  = {en}
}
@unpublished{KoppitzMusunthia2012,
  author    = {Koppitz, J{\"o}rg and Musunthia, Tiwadee},
  title     = {Maximal subsemigroups containing a particular semigroup},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57465},
  year      = {2012},
  abstract  = {We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X = N of natural numbers containing a given subsemigroup W of T(X), where each element of a given set U is a generator of T(X) modulo W. This note continues the study of maximal subsemigroups of the monoid of all full transformations on an infinite set.},
  language  = {en}
}
@phdthesis{Murr2012,
  author    = {Murr, R{\"u}diger},
  title     = {Reciprocal classes of Markov processes : an approach with duality formulae},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-62091},
  school      = {Universit{\"a}t Potsdam},
  year      = {2012},
  abstract  = {This work is concerned with the characterization of certain classes of stochastic processes via duality formulae. In particular we consider reciprocal processes with jumps, a subject up to now neglected in the literature. In the first part we introduce a new formulation of a characterization of processes with independent increments. This characterization is based on a duality formula satisfied by processes with infinitely divisible increments, in particular L{\´e}vy processes, which is well known in Malliavin calculus. We obtain two new methods to prove this duality formula, which are not based on the chaos decomposition of the space of square-integrable function- als. One of these methods uses a formula of partial integration that characterizes infinitely divisible random vectors. In this context, our characterization is a generalization of Stein's lemma for Gaussian random variables and Chen's lemma for Poisson random variables. The generality of our approach permits us to derive a characterization of infinitely divisible random measures. The second part of this work focuses on the study of the reciprocal classes of Markov processes with and without jumps and their characterization. We start with a resume of already existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. Thus we are able to connect the results of characterizations via duality formulae with the theory of stochastic mechanics by our interpretation, and to stochastic optimal control theory by the mathematical approach. As an application we are able to prove an invariance property of the reciprocal class of a Brownian diffusion under time reversal. In the context of pure jump processes we derive the following new results. We describe the reciprocal classes of Markov counting processes, also called unit jump processes, and obtain a characterization of the associated reciprocal class via a duality formula. This formula contains as key terms a stochastic derivative, a compensated stochastic integral and an invariant of the reciprocal class. Moreover we present an interpretation of the characterization of a reciprocal class in the context of stochastic optimal control of unit jump processes. As a further application we show that the reciprocal class of a Markov counting process has an invariance property under time reversal. Some of these results are extendable to the setting of pure jump processes, that is, we admit different jump-sizes. In particular, we show that the reciprocal classes of Markov jump processes can be compared using reciprocal invariants. A characterization of the reciprocal class of compound Poisson processes via a duality formula is possible under the assumption that the jump-sizes of the process are incommensurable.},
  language  = {en}
}
@unpublished{Murr2012,
  author    = {Murr, R{\"u}diger},
  title     = {Reciprocal classes of Markov processes : an approach with duality formulae},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63018},
  year      = {2012},
  abstract  = {In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.},
  language  = {en}
}
@book{Zhuchok2018,
  author    = {Zhuchok, Anatolii V.},
  title     = {Relatively free doppelsemigroups},
  number    = {5},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-427-2},
  issn      = {2199-4951},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-407719},
  publisher      = {Universit{\"a}t Potsdam},
  pages     = {vii, 78},
  year      = {2018},
  abstract  = {A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. We consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as interassociative semigroups, restrictive bisemigroups, dimonoids, and trioids. In the lecture notes numerous examples of doppelsemigroups and of strong doppelsemigroups are given. The independence of axioms of a strong doppelsemigroup is established. A free product in the variety of doppelsemigroups is presented. We also construct a free (strong) doppelsemigroup, a free commutative (strong) doppelsemigroup, a free n-nilpotent (strong) doppelsemigroup, a free n-dinilpotent (strong) doppelsemigroup, and a free left n-dinilpotent doppelsemigroup. Moreover, the least commutative congruence, the least n-nilpotent congruence, the least n-dinilpotent congruence on a free (strong) doppelsemigroup and the least left n-dinilpotent congruence on a free doppelsemigroup are characterized. The book addresses graduate students, post-graduate students, researchers in algebra and interested readers.},
  language  = {en}
}
@unpublished{AntonioukKiselevStepanenkoetal.2012,
  author    = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg and Stepanenko, Vitaly and Tarkhanov, Nikolai Nikolaevich},
  title     = {Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61987},
  year      = {2012},
  abstract  = {The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.},
  language  = {en}
}
@unpublished{HoegeleRuffino2013,
  author    = {H{\"o}gele, Michael and Ruffino, Paulo},
  title     = {Averaging along L{\´e}vy diffusions in foliated spaces},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64926},
  year      = {2013},
  abstract  = {We consider an SDE driven by a L{\´e}vy noise on a foliated manifold, whose trajectories stay on compact leaves. We determine the effective behavior of the system subject to a small smooth transversal perturbation of positive order epsilon. More precisely, we show that the average of the transversal component of the SDE converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to the invariant measures on the leaves (of the unpertubed system) as epsilon goes to 0. In particular we give upper bounds for the rates of convergence. The main results which are proved for pure jump L{\´e}vy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to L{\´e}vy driven SDEs of Marcus type.},
  language  = {en}
}
@unpublished{ShlapunovTarkhanov2012,
  author    = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich},
  title     = {On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators},
  issn      = {2193-6943},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57759},
  year      = {2012},
  abstract  = {We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.},
  language  = {en}
}
@unpublished{Nehring2012,
  author    = {Nehring, Benjamin},
  title     = {Construction of point processes for classical and quantum gases},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59648},
  year      = {2012},
  abstract  = {We propose a new construction of point processes, which generalizes the class of infinitely divisible point processes. Examples are the quantum Boson and Fermion gases as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.},
  language  = {en}
}
@unpublished{BaerBallmann2012,
  author    = {B{\"a}r, Christian and Ballmann, Werner},
  title     = {Boundary value problems for elliptic differential operators of first order},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60023},
  year      = {2012},
  abstract  = {We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.},
  language  = {en}
}