## 510 Mathematik

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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.

The Cauchy problem for the linearised Einstein equation and the Goursat problem for wave equations
(2017)

In this thesis, we study two initial value problems arising in general relativity. The first is the Cauchy problem for the linearised Einstein equation on general globally hyperbolic spacetimes, with smooth and distributional initial data. We extend well-known results by showing that given a solution to the linearised constraint equations of arbitrary real Sobolev regularity, there is a globally defined solution, which is unique up to addition of gauge solutions. Two solutions are considered equivalent if they differ by a gauge solution. Our main result is that the equivalence class of solutions depends continuously on the corre- sponding equivalence class of initial data. We also solve the linearised constraint equations in certain cases and show that there exist arbitrarily irregular (non-gauge) solutions to the linearised Einstein equation on Minkowski spacetime and Kasner spacetime.
In the second part, we study the Goursat problem (the characteristic Cauchy problem) for wave equations. We specify initial data on a smooth compact Cauchy horizon, which is a lightlike hypersurface. This problem has not been studied much, since it is an initial value problem on a non-globally hyperbolic spacetime. Our main result is that given a smooth function on a non-empty, smooth, compact, totally geodesic and non-degenerate Cauchy horizon and a so called admissible linear wave equation, there exists a unique solution that is defined on the globally hyperbolic region and restricts to the given function on the Cauchy horizon. Moreover, the solution depends continuously on the initial data. A linear wave equation is called admissible if the first order part satisfies a certain condition on the Cauchy horizon, for example if it vanishes. Interestingly, both existence of solution and uniqueness are false for general wave equations, as examples show. If we drop the non-degeneracy assumption, examples show that existence of solution fails even for the simplest wave equation. The proof requires precise energy estimates for the wave equation close to the Cauchy horizon. In case the Ricci curvature vanishes on the Cauchy horizon, we show that the energy estimates are strong enough to prove local existence and uniqueness for a class of non-linear wave equations. Our results apply in particular to the Taub-NUT spacetime and the Misner spacetime. It has recently been shown that compact Cauchy horizons in spacetimes satisfying the null energy condition are necessarily smooth and totally geodesic. Our results therefore apply if the spacetime satisfies the null energy condition and the Cauchy horizon is compact and non-degenerate.

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.

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.