## 510 Mathematik

### Refine

#### Year of publication

#### Document Type

- Preprint (374)
- Doctoral Thesis (45)
- Postprint (23)
- Monograph/Edited Volume (8)
- Article (6)
- Part of a Book (3)
- Conference Proceeding (2)
- Master's Thesis (2)

#### Language

- English (430)
- German (29)
- French (3)
- Multiple languages (1)

#### Keywords

- index (11)
- boundary value problems (9)
- elliptic operators (9)
- Fredholm property (8)
- cluster expansion (8)
- K-theory (7)
- pseudodifferential operators (7)
- manifolds with singularities (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)

#### Institute

- Institut für Mathematik (450)
- Extern (13)
- Institut für Philosophie (3)
- Historisches Institut (2)
- Institut für Biochemie und Biologie (2)
- Institut für Informatik und Computational Science (2)
- Interdisziplinäres Zentrum für Dynamik komplexer Systeme (2)
- Präsident | Vizepräsidenten (2)
- Strukturbereich Kognitionswissenschaften (2)
- Institut für Physik und Astronomie (1)

Integral Fourier operators
(2017)

This volume of contributions based on lectures delivered at a school on Fourier Integral Operators
held in Ouagadougou, Burkina Faso, 14–26 September 2015, provides an introduction to Fourier Integral Operators (FIO) for a readership of Master and PhD students as well as any interested layperson. Considering the wide
spectrum of their applications and the richness of the mathematical tools they involve, FIOs lie the cross-road of many a field. This volume offers
the necessary background, whether analytic or geometric, to get acquainted with FIOs, complemented by more advanced material presenting various aspects of active research in that area.

The interdisciplinary workshop STOCHASTIC PROCESSES WITH APPLICATIONS IN THE NATURAL SCIENCES was held in Bogotá, 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 Mathématiques, 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.

In this thesis, stochastic dynamics modelling collective motions of populations, one of the most mysterious type of biological phenomena, are considered. For a system of N particle-like individuals, two kinds of asymptotic behaviours are studied : ergodicity and flocking properties, in long time, and propagation of chaos, when the number N of agents goes to infinity. Cucker and Smale, deterministic, mean-field kinetic model for a population without a hierarchical structure is the starting point of our journey : the first two chapters are dedicated to the understanding of various stochastic dynamics it inspires, with random noise added in different ways. The third chapter, an attempt to improve those results, is built upon the cluster expansion method, a technique from statistical mechanics. Exponential ergodicity is obtained for a class of non-Markovian process with non-regular drift. In the final part, the focus shifts onto a stochastic system of interacting particles derived from Keller and Segel 2-D parabolicelliptic model for chemotaxis. Existence and weak uniqueness are proven.

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.

Raum und Form
(2017)

The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.

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 Lamé equations within the context of elliptic quasicomplexes on compact manifolds with boundary. The non-stationary Lamé 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.

During the drug discovery & development process, several phases encompassing a number of preclinical and clinical studies have to be successfully passed to demonstrate safety and efficacy of a new drug candidate. As part of these studies, the characterization of the drug's pharmacokinetics (PK) is an important aspect, since the PK is assumed to strongly impact safety and efficacy. To this end, drug concentrations are measured repeatedly over time in a study population. The objectives of such studies are to describe the typical PK time-course and the associated variability between subjects. Furthermore, underlying sources significantly contributing to this variability, e.g. the use of comedication, should be identified. The most commonly used statistical framework to analyse repeated measurement data is the nonlinear mixed effect (NLME) approach. At the same time, ample knowledge about the drug's properties already exists and has been accumulating during the discovery & development process: Before any drug is tested in humans, detailed knowledge about the PK in different animal species has to be collected. This drug-specific knowledge and general knowledge about the species' physiology is exploited in mechanistic physiological based PK (PBPK) modeling approaches -it is, however, ignored in the classical NLME modeling approach.
Mechanistic physiological based models aim to incorporate relevant and known physiological processes which contribute to the overlying process of interest. In comparison to data--driven models they are usually more complex from a mathematical perspective. For example, in many situations, the number of model parameters outrange the number of measurements and thus reliable parameter estimation becomes more complex and partly impossible. As a consequence, the integration of powerful mathematical estimation approaches like the NLME modeling approach -which is widely used in data-driven modeling -and the mechanistic modeling approach is not well established; the observed data is rather used as a confirming instead of a model informing and building input.
Another aggravating circumstance of an integrated approach is the inaccessibility to the details of the NLME methodology so that these approaches can be adapted to the specifics and needs of mechanistic modeling. Despite the fact that the NLME modeling approach exists for several decades, details of the mathematical methodology is scattered around a wide range of literature and a comprehensive, rigorous derivation is lacking. Available literature usually only covers selected parts of the mathematical methodology. Sometimes, important steps are not described or are only heuristically motivated, e.g. the iterative algorithm to finally determine the parameter estimates.
Thus, in the present thesis the mathematical methodology of NLME modeling is systemically described and complemented to a comprehensive description,
comprising the common theme from ideas and motivation to the final parameter estimation. Therein, new insights for the interpretation of different approximation methods used in the context of the NLME modeling approach are given and illustrated; furthermore, similarities and differences between them are outlined. Based on these findings, an expectation-maximization (EM) algorithm to determine estimates of a NLME model is described.
Using the EM algorithm and the lumping methodology by Pilari2010, a new approach on how PBPK and NLME modeling can be combined is presented and exemplified for the antibiotic levofloxacin. Therein, the lumping identifies which processes are informed by the available data and the respective model reduction improves the robustness in parameter estimation. Furthermore, it is shown how apriori known factors influencing the variability and apriori known unexplained variability is incorporated to further mechanistically drive the model development. Concludingly, correlation between parameters and between covariates is automatically accounted for due to the mechanistic derivation of the lumping and the covariate relationships.
A useful feature of PBPK models compared to classical data-driven PK models is in the possibility to predict drug concentration within all organs and tissue in the body. Thus, the resulting PBPK model for levofloxacin is used to predict drug concentrations and their variability within soft tissues which are the site of action for levofloxacin. These predictions are compared with data of muscle and adipose tissue obtained by microdialysis, which is an invasive technique to measure a proportion of drug in the tissue, allowing to approximate the concentrations in the interstitial fluid of tissues. Because, so far, comparing human in vivo tissue PK and PBPK predictions are not established, a new conceptual framework is derived. The comparison of PBPK model predictions and microdialysis measurements shows an adequate agreement and reveals further strengths of the presented new approach.
We demonstrated how mechanistic PBPK models, which are usually developed in the early stage of drug development, can be used as basis for model building in the analysis of later stages, i.e. in clinical studies. As a consequence, the extensively collected and accumulated knowledge about species and drug are utilized and updated with specific volunteer or patient data. The NLME approach combined with mechanistic modeling reveals new insights for the mechanistic model, for example identification and quantification of variability in mechanistic processes. This represents a further contribution to the learn & confirm paradigm across different stages of drug development.
Finally, the applicability of mechanism--driven model development is demonstrated on an example from the field of Quantitative Psycholinguistics to analyse repeated eye movement data. Our approach gives new insight into the interpretation of these experiments and the processes behind.

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.