TY  - INPR
A1  - Blanchard, Gilles
A1  - Mücke, Nicole
T1  - Optimal rates for regularization of statistical inverse learning problems
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 5 
KW  - statistical inverse problem
KW  - minimax rate
KW  - kernel method
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89782
SN  - 2193-6943
VL  - 5
IS  - 5
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Mera, Azal
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - The Neumann problem after Spencer
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 6 
KW  - elliptic complex
KW  - manifold with boundary
KW  - Hodge theory
KW  - Neumann problem
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-90631
SN  - 2193-6943
VL  - 5
IS  - 6
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Mera, Azal
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Navier-Stokes equations for elliptic complexes
N2  - We continue our study of invariant forms of the classical equations of mathematical physics,
such as the Maxwell equations or the Lamé 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)12 
KW  - Navier-Stokes equations
KW  - classical solution
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-85592
SN  - 2193-6943
VL  - 4
IS  - 12
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Conforti, Giovanni
T1  - Reciprocal classes of continuous time Markov Chains
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 8 
KW  - random walks on graphs
KW  - bridges of random walks
KW  - reciprocal characteristics
KW  - Schrödinger problem
KW  - integration by parts on path space
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78234
SN  - 2193-6943
VL  - 4
IS  - 8
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Gibali, Aviv
A1  - Shoikhet, David
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - On the convergence of continuous Newton method
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)10 
KW  - Newton method
KW  - spirallike function
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-81537
SN  - 2193-6943
VL  - 4
IS  - 10
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Cheng, Yuan
T1  - Recursive state estimation in dynamical systems
Y1  - 2016
ER  - 
TY  - THES
A1  - Mera, Azal Jaafar Musa
T1  - The Navier-Stokes equations for elliptic quasicomplexes
T1  - Die Navier–Stokes–Gleichungen für elliptische Quasikomplexe
N2  - 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.
N2  - Die klassischen Navier–Stokes–Differentialgleichungen  der Hydrodynamik werden in der Regel im Rahmen der Vektoranalysis formuliert. Mehr versprechend ist die Formulierung dieser Gleichungen in Termen von Differentialformen vom Grad 1. Auf diesem Weg beinhaltet die Untersuchung der Navier–Stokes–Gleichungen die Analyse des de Rhamschen  Komplexes. Insbesondere ermöglicht die Hodge–Theorie für den de Rham–Komplex den Druck aus den Gleichungen zu eliminieren. Die Navier–Stokes–Gleichungen bilden ein parabolisches System mit einem nichtlinearen Term, welcher Sinn nur für die Pfaffschen Formen (d.h Formen vom Grad 1) hat. Ein einfacheres Modell für Dynamik der inkompressiblen viskosen Flüssigkeit wird von der Burgers–Gleichungen gegeben. Diese Arbeit richtet sich an das Studium der invarianten Struktur der Navier–Stokes–Gleichungen, die eng mit der algebraischen Struktur des de Rham–Komplexes im schritt 1 zusammen steht. Zu diesem Zweck stellen wir vor die Navier–Stokes–Gleichungen im Zusammenhang mit jedem elliptischen Quasikomplex von Differentialoperatoren der ersten Ordnung. So ähneln die Gleichungen den klassischen Navier–Stokes–Gleichungen, einschließlich allgemeiner Geschwindigkeit– und Druckvektoren. Elimination des Drucks aus den verallgemeinerten Navier–Stokes–Gleichungen gibt eine gute Motivation für die Untersuchung des Neumann–Problems nach Spencer für elliptische Quasikomplexe. Eine solche Untersuchung ist auch in der Arbeit mit der Erörterung der Lamé-Gleichungen im Kontext der elliptischen Quasikomplexe auf kompakten Mannigfaltigkeiten mit Rand. Die nichtstationären Lamé-Gleichungen bilden ein hyperbolisches System. Allerdings gibt die Studie des ersten gemischten Problems für sie eine gute Erfahrung, um die linearisierten Navier–Stokes–Gleichungen anzugreifen. Auf dieser Basis beschreiben wir eine Klasse von nichtlinearen Störungen der Navier–Stokes–Gleichungen, für welche die Lösungsresultate noch gelten.
KW  - Navier-Stokes-Gleichungen
KW  - elliptische Quasi-Komplexe
KW  - Navier-Stoks equations
KW  - elliptic quasicomplexes
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-398495
ER  - 
TY  - CHAP
A1  - Valleriani, Angelo
A1  - Roelly, Sylvie
A1  - Kulik, Alexei Michajlovič
ED  - Roelly, Sylvie
ED  - Högele, Michael
ED  - Rafler, Mathias
T1  - Stochastic processes with applications in the natural sciences
BT  - international workshop at Universidad de los Andes, Bogotá, Colombia
T2  - Lectures in pure and applied mathematics
N2  - 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.
T3  - Lectures in pure and applied mathematics - 4 
KW  - macromolecular decay
KW  - Markov processes
KW  - branching processes
KW  - long-time behaviour
KW  - makromolekularer Zerfall
KW  - Markovprozesse
KW  - Verzweigungsprozesse
KW  - Langzeitverhalten
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-401802
SN  - 978-3-86956-414-2
SN  - 2199-4951
SN  - 2199-496X
IS  - 4
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Mücke, Nicole
T1  - Direct and inverse problems in machine learning
T1  - Direkte und inverse Probleme im maschinellen Lernen
BT  - kernel methods and spectral regularization
BT  - Kern Methoden und spektrale Regularisierung
N2  - 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.
N2  - In dieser Arbeit analysieren wir ein zufälliges und verrauschtes inverses Regressionsmodell im random design. Wir konstruiueren aus gegebenen Daten eine Schätzung der unbekannten Funktion, von der wir annehmen, dass sie in einem Hilbertraum mit reproduzierendem Kern liegt.

Ein erstes Hauptergebnis dieser Arbeit betrifft obere Schranken an die Konvergenzraten. Wir legen sog. source conditions fest, definiert über geeignete Kugeln im Wertebereich von (reellen) Potenzen des normierten Kern-Kovarianzoperators. Das führt zu  einer Einschränkung der Klasse der Verteilungen  in einem statistischen Modell, in dem die spektrale Asymptotik des von der Randverteilung abhängigen Kovarianzoperators eingeschränkt wird.

In diesem Kontext zeigen wir obere und entsprechende untere Schranken für die Konvergenzraten für eine sehr allgemeine Klasse spektraler Regularisierungsmethoden und etablieren damit die sog.  Minimax-Optimalität dieser Raten. Da selbst bei optimalen Konvergenzraten Kernmethoden, angewandt auf große Datenmengen, noch unbefriedigend viel Zeit verschlingen und hohen Speicherbedarf aufweisen, untersuchen wir einen Zugang zur Zeitersparnis und zur Reduktion des Speicherbedarfs detaillierter. Wir studieren das sog.  distributed learning  und beweisen für unsere Klasse allgemeiner spektraler Regularisierungen ein neues Resultat, allerdings immer noch unter der Annahme einer bekannten a priori Regularität der Zielfunktion, ausgedrückt durch die Fixierung einer source condition. Das große Problem bei der Behandlung realer Daten ist das der  Adaptivität, d.h. die Angabe eines Verfahrens, das ohne eine solche  a priori Voraussetzung  einen in einem gewissen Sinn optimalen Schätzer aus den  Daten konstruiert. Das behandeln wir vermöge einer Variante des Balancing principle.
KW  - inverse problems
KW  - kernel methods
KW  - minimax optimality
KW  - inverse Probleme
KW  - Kern Methoden
KW  - Minimax Optimalität
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-403479
ER  - 
TY  - INPR
A1  - Polkovnikov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Riemann-Hilbert problem for the Moisil-Teodorescu system
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 3 
KW  - Dirac operator
KW  - Riemann-Hilbert problem
KW  - Fredholm operators
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-397036
VL  - 6
IS  - 3
ER  - 
TY  - THES
A1  - Jende, Alexander
T1  - On the characterization of particular orthogroups by disjunctions of identities
N2  - 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.
N2  - In dieser Dissertation wird die Charakterisierung von Orthogruppen durch sogenannte Alternatividentitäten diskutiert. Die Orthogruppen bilden eine Unterklasse der Klasse der vollständig regulären Halbgruppen, einer Verallgemeinerung des Konzeptes einer Gruppe. Somit besitzen alle Elemente einer Orthogruppe eine Art inverses Element, mit welchem sie sogar kommutieren. Nach einem elementaren Ergebnis von A.H. Clifford lässt sich jede vollständig reguläre Halbgruppe als Halbverband von vollständig einfachen Halbgruppen darstellen. Dies erlaubt eine grobe Beschreibung der Struktur einer solchen Halbgruppe. Insbesondere lässt sich jede Orthogruppe als Halbverband von rektangulären Gruppen, welche isomorph zu direkten Produkten von rektangulären Bändern und Gruppen sind, beschreiben. Halbverbände rektangulärer Gruppen verschiedener Klassen werden unter Verwendung des Konzeptes der Alternativvarietät, einer Verallgemeinerung des klassischen Begriffs der Varietät nach Birkhoff, charakterisiert. Nach grundlegenden Definitionen und Aussagen zu Halbgruppen, werden die Alternatividentitäten eingeführt und notwendige theoretische Aspekte erläutert. Insbesondere wird eine Alternatividentität angegeben, welche eine hinreichende Bedingung dafür ist, dass eine Halbgruppe vollständig regulär ist. Weiterhin werden in diesem Kontext Aussagen zu Rees-Matrix-Halbgruppen, einer möglichen Darstellung der vollständig einfachen Halbgruppen, hergeleitet. Ein Hauptresultat der Dissertation ist die allgemeine Beschreibung von Alternatividentitäten, sodass eine vollständig reguläre Halbgruppe ein Halbverband von Linksgruppen (Rechtsgruppen / Gruppen) ist. In diesem Fall ist die vollständig reguläre Halbgruppe sogar eine Orthogruppe. Weiterhin werden verschiedene Klassen von rektangulären Gruppen definiert, die einen bestimmten Exponenten aus einer gegebenen Menge von paarweise teilerfremden natürlichen Zahlen besitzen. Ein wichtiges Ergebnis ist die Charakterisierung der Klasse aller Halbverbände von bestimmten rektangulären Gruppen (aus den vorher definierten Klassen) mittels einer mengentheoretisch minimalen Menge von Alternatividentitäten. Es werden außerdem Halbverbände von Gruppen (sogenannte Clifford-Halbgruppen) näher untersucht. Hierfür werden abelsche Gruppen mit bestimmtem Exponenten betrachtet und teilweise bekannte Resultate aus der Theorie der Clifford-Halbgruppen mittels des Konzeptes der Alternatividentitäten auf alternativem Weg bewiesen. Als konkrete Anwendung der Ergebnisse über Halbverbände von Linksnullhalbgruppen und Rechtsnullhalbgruppen wird die Identifizierung einer speziellen Transformationshalbgruppe dargelegt. Um detailliertere Aussagen über das Produkt zweier beliebiger Elemente eines Halbverbandes von Halbgruppen zu erhalten, wird das Konzept der starken Halbverbände von Halbgruppen eingeführt. Hierbei ist bekannt, dass jeder Halbverband von Gruppen ein starker Halbverband von Gruppen ist. Somit lässt sich ein starker Halbverband von Gruppen mit bestimmten, paarweise teilerfremden Exponenten durch Alternatividentitäten charakterisieren. Zusätzlich wird die Klasse aller starken Halbverbände von Linksnullhalbgruppen und Rechtsnullhalbgruppen durch diese Art von Identitäten beschrieben und in die Theorie der normalen Bänder eingeordnet. Eine mögliche Ergänzung der beschriebenen Halbverbände von rektangulären Gruppen erfolgt durch die zusätzliche Forderung einer totalen Ordnung (im Sinne von Ketten von Halbgruppen). Hierfür wird ebenfalls eine entsprechende Charakterisierung durch Alternatividentitäten präsentiert, welche offensichtlich minimal ist. Eine Auflistung offener Fragen, welche während der Arbeit an der Dissertation entstanden sind, jedoch unbearbeitet bleiben mussten, ist beigefügt.
KW  - Halbgruppentheorie
KW  - Orthogruppen
KW  - Alternatividentitäten
KW  - Alternativvarietäten
KW  - Clifford-Halbgruppen
KW  - starker Halbverband von Halbgruppen
KW  - Kette von Halbgruppen
KW  - semigroup theory
KW  - orthogroup
KW  - disjunction of identities
KW  - alternative variety
KW  - Clifford semigroup
KW  - strong semilattice of semigroups
KW  - chain of semigroups
Y1  - 2018
ER  - 
TY  - THES
A1  - Tinpun, Kittisak
T1  - Relative rank of infinite full transformation semigroups with restricted range
Y1  - 2019
ER  - 
TY  - INPR
A1  - Ly, I.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - The cauchy problem for nonlinear elliptic equations
N2  - This paper is devoted to investigation of the Cauchy problem for nonlinear elliptic equations with a small parameter.
T3  - Preprint - (2007) 01 
Y1  - 2007
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30228
ER  - 
TY  - THES
A1  - Böhne, Sebastian
T1  - Different degrees of formality
T1  - Verschiedene Formalitätsgrade
BT  - an introduction to the concept and a demonstration of its usefulness
BT  - Vorstellung des Konzepts und Nachweis seiner Nützlichkeit
N2  - 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.
N2  - In dieser Dissertation stellen wir das Konzept der Formalitätsgrade vor, welches sich gegen eine dualistische Sichtweise richtet, die nur zwischen formalen und informalen Beweisen unterscheidet. Letztere Sichtweise spiegelt nämlich die Unterschiede zwischen den als informal klassifizierten Argumentationen nicht wieder und ist außerdem unproduktiv, weil sie nicht in der Lage ist, das individuelle Potential der jeweiligen Argumentationsstile wertzuschätzen und auszuschöpfen.

Die Dissertation hat zwei Teile. Im ersten analysieren wir das Konzept der Formalitätsgrade (eine Diskussion über die Vorteile der jeweiligen Grade eingeschlossen), während wir im zweiten Teil die Nützlichkeit der Formalitätsgrade anhand von drei Fallbeispielen nachweisen. Im ersten von diesen werden wir Haskell B. Currys Sichtweise zur Mathematik, die nebenbei bemerkt von größter Wichtigkeit für den ersten Teil der Dissertation ist, mithilfe der verschiedenen Formalitätsgrade reparieren. Im zweiten Fallbeispiel zeigen wir auf, wie die Beachtung der verschiedenen Formalitätsgrade den Studenten dabei helfen kann, das Beweisen zu erlernen. Im letzten Fallbeispiel werden wir dann zeigen, wie die Vorteile von Beweisen verschiedener Formalitätsgrade durch die Anwendung sogenannter Taktiken mittleren Formalitätsgrades kombiniert werden können. Zusammen zeigen die drei Fallbeispiele, dass die Formalitätsgrade eine überzeugende Lösung für das Problem des ungenutzten Potentials darstellen.
KW  - argumentation
KW  - Coq
KW  - Curry
KW  - degree of formality
KW  - formalism
KW  - logic
KW  - mathematics education
KW  - philosophy of mathematics
KW  - proof
KW  - proof assistant
KW  - proof environment
KW  - tactic
KW  - Argumentation
KW  - Beweis
KW  - Beweisassistent
KW  - Beweisumgebung
KW  - Coq
KW  - Curry
KW  - Formalismus
KW  - Formalitätsgrad
KW  - Logik
KW  - Mathematikdidaktik
KW  - Mathematikphilosophie
KW  - Taktik
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-423795
N1  - CCS -> Applied computing -> Education -> Interactive learning environments
CCS -> Theory of computation -> Logic
CCS -> Computing methodologies -> Symbolic and algebraic manipulation -> Symbolic and algebraic algorithms -> Theorem proving algorithms
ER  - 
TY  - JOUR
A1  - Kiy, Alexander
A1  - Dehne, Julian
A1  - Bussler, Dirk
T1  - Aufbau einer Cloud-Speicherlösung und Integration in bestehende IuK-Infrastrukturen am Beispiel ownCloud an der Universität Potsdam
JF  - Cloudspeicher im Hochschuleinsatz 2015: Proceedings der Tagung "Cloudspeicher im Hochschuleinsatz" am 07. und 08. Mai 2015 am IT-Service-Center (tubIT) der Technischen Universität Berlin
N2  - 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.
KW  - Cloud Computing
KW  - Hochschule
KW  - Studium
Y1  - 2015
SN  - 978-3-7983-2780-1
SP  - 61
EP  - 72
PB  - Universitätsverlag der TU Berlin
CY  - Berlin
ER  - 
TY  - CHAP
A1  - Kiy, Alexander
A1  - Strelczuk, Frederik
A1  - Lucke, Ulrike
ED  - Horbach, Matthias
T1  - Ein Plugin zur integrierten Literaturverwaltung in Moodle
T2  - Ein Plugin zur integrierten Literaturverwaltung in Moodle
Y1  - 2013
SN  - 978-3-88579-614-5
SP  - 334
EP  - 345
PB  - Gesellschaft für Informatik e.V.
CY  - Bonn
ER  - 
TY  - THES
A1  - Münch, Florentin
T1  - Discrete Ricci curvature, diameter bounds and rigidity
Y1  - 2019
ER  - 
TY  - JOUR
A1  - Hanisch, Florian
A1  - Ludewig, Matthias
T1  - A rigorous construction of the supersymmetric path integral associated to a compact spin manifold
JF  - Communications in mathematical physics
N2  - 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.
Y1  - 2022
U6  - https://doi.org/10.1007/s00220-022-04336-7
SN  - 0010-3616
SN  - 1432-0916
VL  - 391
IS  - 3
SP  - 1209
EP  - 1239
PB  - Springer
CY  - Berlin ; Heidelberg
ER  - 
TY  - JOUR
A1  - Seelig, Stefan A.
A1  - Rabe, Maximilian Michael
A1  - Malem-Shinitski, Noa
A1  - Risse, Sarah
A1  - Reich, Sebastian
A1  - Engbert, Ralf
T1  - Bayesian parameter estimation for the SWIFT model of eye-movement control during reading
JF  - Journal of mathematical psychology
N2  - 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.
KW  - dynamical models
KW  - reading
KW  - eye movements
KW  - saccades
KW  - likelihood function
KW  - Bayesian inference
KW  - MCMC
KW  - interindividual differences
Y1  - 2020
U6  - https://doi.org/10.1016/j.jmp.2019.102313
SN  - 0022-2496
SN  - 1096-0880
VL  - 95
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Pikovskij, Arkadij
T1  - Chimeras on a social-type network
JF  - Mathematical modelling of natural phenomena : MMNP
N2  - 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.
KW  - Network
KW  - Chimera
KW  - correlations
KW  - Lyapunov exponent
Y1  - 2021
U6  - https://doi.org/10.1051/mmnp/2021012
SN  - 0973-5348
SN  - 1760-6101
VL  - 16
PB  - EDP Sciences
CY  - Les Ulis
ER  - 
TY  - BOOK
A1  - Pikovskij, Arkadij
A1  - Politi, Antonio
T1  - Lyapunov Exponents
BT  - a tool to explore complex dynamics
N2  - 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.
Y1  - 2016
SN  - 978-1-107-03042-8
PB  - Cambridge University Press
CY  - Cambridge
ER  - 
TY  - GEN
A1  - Straube, Arthur V.
A1  - Pikovskij, Arkadij
T1  - Pattern formation induced by time-dependent advection
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 575 
KW  - pattern formation
KW  - reaction-advection-diffusion equation
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413140
SN  - 1866-8372
IS  - 575
SP  - 138-147
ER  - 
TY  - JOUR
A1  - De Oliveira Gomes, André
A1  - Högele, Michael Anton
T1  - The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps
JF  - Stochastics and dynamics
N2  - 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.
KW  - Freidlin-Wentzell theory
KW  - large deviations principle
KW  - accelerated small
KW  - noise Levy diffusions
KW  - first passage times
KW  - first exit location
KW  - strongly tempered stable Levy measure
Y1  - 2021
U6  - https://doi.org/10.1142/S0219493721500192
SN  - 0219-4937
SN  - 1793-6799
VL  - 21
IS  - 04
PB  - World Scientific
CY  - Singapore
ER  - 
TY  - JOUR
A1  - Wiljes, Jana de
A1  - Tong, Xin T.
T1  - Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations
JF  - Nonlinearity
N2  - 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.
KW  - data assimilation
KW  - stability and accuracy
KW  - dimension independent bound
KW  - localisation
KW  - high dimensional
KW  - filter
KW  - nonlinear
Y1  - 2020
U6  - https://doi.org/10.1088/1361-6544/ab8d14
SN  - 0951-7715
SN  - 1361-6544
VL  - 33
IS  - 9
SP  - 4752
EP  - 4782
PB  - IOP Publ.
CY  - Bristol
ER  - 
TY  - THES
A1  - Hanisch, Florian
T1  - Variational problems on supermanifolds
T1  - Variationsprobleme auf Supermannigfaltigkeiten
N2  - 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.
N2  - In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt.
KW  - Supergeometrie
KW  - Variationsrechnung
KW  - Differentialoperatoren
KW  - Funktorgeometrie
KW  - supergeometry
KW  - variational calculus
KW  - differential operators
KW  - functor geometry
Y1  - 2011
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59757
ER  - 
TY  - JOUR
A1  - Garmendia, Alfonso
A1  - Zambon, Marco
T1  - Quotients of singular foliations and Lie 2-group actions
JF  - Journal of noncommutative geometry
N2  - 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.
KW  - Lie groupoid
KW  - singular foliation
KW  - fibration
Y1  - 2021
U6  - https://doi.org/10.4171/JNCG/434
SN  - 1661-6952
SN  - 1661-6960
VL  - 15
IS  - 4
SP  - 1251
EP  - 1283
PB  - EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut für Mathematik, Technische Universität Berlin
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Schick, Thomas
A1  - Seyedhosseini, Mehran
T1  - On an index theorem of Chang, Weinberger and Yu
JF  - Münster journal of mathematics
N2  - 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.
Y1  - 2021
U6  - https://doi.org/10.17879/59019522628
SN  - 1867-5778
SN  - 1867-5786
VL  - 14
IS  - 1
SP  - 123
EP  - 154
PB  - WWU, Fachbereich Mathematik und Informatik
CY  - Münster
ER  - 
TY  - JOUR
A1  - Tomovski, Živorad
A1  - Metzler, Ralf
A1  - Gerhold, Stefan
T1  - Fractional characteristic functions, and a fractional calculus approach for moments of random variables
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - 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.
KW  - Fractional calculus (primary)
KW  - Characteristic function
KW  - Mittag-Leffler
KW  - function
KW  - Fractional moments
KW  - Mellin transform
Y1  - 2022
U6  - https://doi.org/10.1007/s13540-022-00047-x
SN  - 1314-2224
VL  - 25
IS  - 4
SP  - 1307
EP  - 1323
PB  - De Gruyter
CY  - Berlin ; Boston
ER  - 
TY  - JOUR
A1  - Purinton, Benjamin
A1  - Bookhagen, Bodo
T1  - Multiband (X, C, L) radar amplitude analysis for a mixed sand- and gravel-bed river in the eastern Central Andes
JF  - Remote sensing of environment : an interdisciplinary journal
N2  - 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.
KW  - SAR amplitude
KW  - Radar backscatter
KW  - Surface roughness
KW  - Fluvial
KW  - geomorphology
KW  - TerraSAR-X/TanDEM-X
KW  - Sentinel-1
KW  - ALOS-2 PALSAR-2
Y1  - 2020
U6  - https://doi.org/10.1016/j.rse.2020.111799
SN  - 0034-4257
SN  - 1879-0704
VL  - 246
PB  - Elsevier
CY  - New York
ER  - 
TY  - JOUR
A1  - Kwarikunda, Diana
A1  - Schiefele, Ulrich
A1  - Ssenyonga, Joseph
A1  - Muwonge, Charles Magoba
T1  - Secondary school students’ motivation profiles for physics learning
BT  - Relations with cognitive learning strategies, gender, attitudes and individual interest
JF  - 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
N2  - 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.
KW  - Motivation profiles
KW  - physics learning
KW  - latent profile analysis
Y1  - 2021
U6  - https://doi.org/10.1080/18117295.2021.1956720
SN  - 1028-8457
VL  - 25
IS  - 2
SP  - 197
EP  - 210
PB  - Routledge, Taylor & Francis Group
CY  - Abingdon
ER  - 
TY  - INPR
A1  - Dereudre, David
A1  - Mazzonetto, Sara
A1  - Roelly, Sylvie
T1  - Exact simulation of Brownian diffusions with drift admitting jumps
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 7 
KW  - exact simulation method
KW  - skew Brownian motion
KW  - skew diffusion
KW  - Brownian motion with discontinuous drift
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-91049
SN  - 2193-6943
VL  - 5
IS  - 7
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - GEN
A1  - Roelly, Sylvie
A1  - Dereudre, David
T1  - Propagation of Gibbsiannes for infinite-dimensional gradient Brownian diffusions
N2  - 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.
KW  - infinite-dimensional Brownian diffusion
KW  - Gibbs measure
KW  - cluster expansion
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6918
ER  - 
TY  - INPR
A1  - Dereudre, David
A1  - Roelly, Sylvie
T1  - Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions
N2  - 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.
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 06 
KW  - infinite-dimensional Brownian diffusion
KW  - Gibbs measure
KW  - cluster expansion
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51535
ER  - 
TY  - INPR
A1  - Dereudre, David
A1  - Mazzonetto, Sara
A1  - Roelly, Sylvie
T1  - An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 9 
KW  - skew Brownian motion
KW  - semipermeable barriers
KW  - distorted Brownian motion
KW  - local time
KW  - rejection sampling
KW  - exact simulation
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-80613
SN  - 2193-6943
VL  - 4
IS  - 9
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Dereudre, David
A1  - Roelly, Sylvie
T1  - Path-dependent infinite-dimensional SDE with non-regular drift : an existence result
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)11 
KW  - Infinite-dimensional SDE
KW  - non-Markov drift
KW  - non-regular drift
KW  - variational principle
KW  - specific entropy
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72084
SN  - 2193-6943
VL  - 3
IS  - 11
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - GEN
A1  - Roelly, Sylvie
A1  - Dereudre, David
T1  - On Gibbsianness of infinite-dimensional diffusions
N2  - 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
KW  - infinite-dimensional Brownian diffusion
KW  - Gibbs field
KW  - cluster expansion
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6692
ER  - 
TY  - JOUR
A1  - Dereudre, David
A1  - Roelly, Sylvie
T1  - Path-dependent infinite-dimensional SDE with non-regular drift
BT  - an existence result
JF  - Annales de l'Institut Henri Poincaré : B, Probability and statistics
N2  - 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.
N2  - Nous établissons, dans cet article, l’existence de solutions faibles pour un système infini-dimensionnel de diffusions browniennes. Le terme de dérive est véritablement général, au sens où il est supposé n’être ni borné, ni continu, ni Markovien. Nous supposons cependant que la loi initiale admet une entropie spécifique finie.
L’originalité de notre méthode consiste en l’utilisation de la bornitude de l’entropie spécifique comme critère de tension et en l’identification des solutions du système comme solutions d’un problème variationnel sur l’espace des trajectoires. Notre résultat améliore clairement ceux préexistants concernant des dynamiques libres perturbées par des dérives bornées.
KW  - Infinite-dimensional SDE
KW  - Non-Markov drift
KW  - Non-regular drift
KW  - Variational principle
KW  - Specific entropy
Y1  - 2017
U6  - https://doi.org/10.1214/15-AIHP728
SN  - 0246-0203
VL  - 53
IS  - 2
SP  - 641
EP  - 657
PB  - Inst. of Mathematical Statistics
CY  - Bethesda
ER  - 
TY  - THES
A1  - Mazzonetto, Sara
T1  - On the exact simulation of (skew) Brownian diffusions with discontinuous drift
T1  - Über die exakte Simulation (skew) Brownsche Diffusionen mit unstetiger Drift
T1  - Simulation exacte de diffusions browniennes (biaisées) avec dérive discontinue
N2  - 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 Itô 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.
N2  - In dieser Dissertation wird die exakte Simulation zweier Klassen reeller Brownscher Diffusionen untersucht: die multi-skew Brownsche Bewegung mit konstanter Drift sowie die Brownsche Diffusionen mit einer Drift mit endlich vielen Sprüngen.
Die skew Brownsche Bewegung wurde in den sechzigern Jahren von Itô and McKean als eine Brownsche Bewegung eingeführt, für die die Richtung ihrer Exkursionen am Ursprung zufällig mit einer gegebenen Wahrscheinlichkeit ausgewürfelt wird. Solche asymmetrischen Prozesse verhalten sich im Wesentlichen wie der Originalprozess außer bei 0, das sich wie eine semipermeable Barriere verhält. Allgemeiner sind skew Diffusionsprozesse mit mehreren semipermeablen Barrieren, auch multi-skew Diffusionen genannt, Diffusionsprozesse mit Ausnahme an den Barrieren, wo sie jeweils teilweise reflektiert wird. Natürlich ist eine multi-skew Diffusion durch eine stochastische Differentialgleichung mit Lokalzeiten (diese bewirken die Semipermeabilität) oder durch ihren infinitesimalen Generator als Markov Prozess charakterisiert. 
In dieser Arbeit leiten wir zunächst eine Konturintegraldarstellung der Übergangshalbgruppe der multi-skew Brownschen Bewegung mit konstanter Drift durch eine feine Analyse ihrer komplexen Eigenschaften her. Dank dieser Darstellung  wird eine explizite Darstellung der Übergangswahrscheinlichkeiten der zweifach-skew Brownschen Bewegung mit konstanter Drift als eine unendliche Reihe Gaußscher Dichten erhalten.
Anschlieẞend wird eine nützliche Verallgemeinerung der bekannten Verwerfungsmethode vorgestellt. Dieses grundlegende Verfahren ermöglicht Realisierungen von Zufallsvariablen, sobald man eine leicht zu simulierende Zufallsvariable derart findet, dass der Quotient der Dichten beider Zufallsvariablen beschränkt ist. Die verallgmeinerte Verwerfungsmethode erlaubt eine exakte Simulation für Dichten, die nur approximiert werden können. Die Originalität unseres Verfahrens liegt nun darin, dass wir, abgesehen von der rechnerbedingten Approximation, exakt von der Verteilung ohne Approximation simulieren.
In einer Anwendung simulieren wir die zweifach-skew Brownsche Bewegung mit oder ohne konstanter Drift. Die Ausgangsdichte ist dabei die der Brownschen Bewegung mit konstanter Drift, und wir geben gleichmäẞige Schranken des Quotienten der Dichten an. Dazu werden numerische Simulationen gezeigt, um die Leistungsfähigkeit des Verfahrens zu demonstrieren.
Das zweite Ziel dieser Arbeit ist die Entwicklung eines exakten Simulationsverfahrens für Brownsche Diffusionen, deren Drift mehrere Sprünge hat. In der Literatur wurden bisher nur Diffusionen mit stetiger Drift bzw. mit einer Drift mit höchstens einem Sprung behandelt. Unser Verfahren erlaubt den Umgang mit jeder endlichen Anzahl von Sprüngen. Insbesondere wird der Fall zweier Sprünge behandelt, da unser Simulationsverfahren mit den bereits erhaltenen Übergangswahrscheinlichkeiten der zweifach-skew Brownschen Bewegung verwandt ist. An mehreren Beispielen demonstrieren wir die Effizienz unseres Ansatzes.
KW  - exact simulation
KW  - exakte Simulation
KW  - skew diffusions
KW  - Skew Diffusionen
KW  - local time
KW  - discontinuous drift
KW  - diskontinuierliche Drift
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102399
ER  - 
TY  - JOUR
A1  - Kempton, Mark
A1  - Münch, Florentin
A1  - Yau, Shing-Tung
T1  - A homology vanishing theorem for graphs with positive curvature
JF  - Communications in analysis and geometry
N2  - We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers.
Y1  - 2021
UR  - https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2021/0029/0006/CAG-2021-0029-0006-a005.pdf
U6  - https://doi.org/10.4310/CAG.2021.v29.n6.a5
SN  - 1019-8385
SN  - 1944-9992
VL  - 29
IS  - 6
SP  - 1449
EP  - 1473
PB  - International Press of Boston
CY  - Somerville
ER  - 
TY  - GEN
A1  - Butuzov, Valentin F.
A1  - Nefedov, N. N.
A1  - Recke, Lutz
A1  - Omel'chenko, Oleh
T1  - Partly dissipative system with multizonal initial and boundary layers
T2  - Journal of Physics: Conference Series
N2  - 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.
Y1  - 2019
U6  - https://doi.org/10.1088/1742-6596/1205/1/012009
SN  - 1742-6588
SN  - 1742-6596
VL  - 1205
PB  - IOP Publ.
CY  - Bristol
ER  - 
TY  - GEN
A1  - Omel'chenko, Oleh
T1  - Travelling chimera states in systems of phase oscillators with asymmetric nonlocal coupling
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1169 
KW  - chimera states
KW  - nonlocally coupled phase oscillators
KW  - Ott–Antonsen equation
KW  - forced symmetry breaking
KW  - travelling waves
KW  - continuation
KW  - stability
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-518141
SN  - 1866-8372
IS  - 2
SP  - 611
EP  - 642
ER  - 
TY  - JOUR
A1  - Omel'chenko, Oleh
T1  - Mathematical framework for breathing chimera states
JF  - Journal of nonlinear science
N2  - 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.
KW  - Coupled oscillators
KW  - Breathing chimera states
KW  - Coherence-incoherence
KW  - patterns
KW  - Ott-Antonsen equation
KW  - Periodic solutions
KW  - Stability
Y1  - 2022
U6  - https://doi.org/10.1007/s00332-021-09779-1
SN  - 0938-8974
SN  - 1432-1467
VL  - 32
IS  - 2
PB  - Springer
CY  - New York
ER  - 
TY  - GEN
A1  - Chechkin, Aleksei V.
A1  - Zaid, Irwin M.
A1  - Lomholt, Michael A.
A1  - Sokolov, Igor M.
A1  - Metzler, Ralf
T1  - Bulk-mediated surface diffusion on a cylinder in the fast exchange limit
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 593 
KW  - Bulk-mediated diffusion;
KW  - anomalous diffusion
KW  - Levy flights
KW  - stochastic processes
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-415480
SN  - 1866-8372
IS  - 593
SP  - 114
EP  - 126
ER  - 
TY  - THES
A1  - Rabe, Maximilian Michael
T1  - Modeling the interaction of sentence processing and eye-movement control in reading
T1  - Modellierung der Interaktion von Satzverarbeitung und Blickbewegungskontrolle beim Lesen
N2  - 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.
N2  - Die Evaluierung prozessorientierter kognitiver Theorien durch zeitlich geordnete Beobachtungen ist ein zentraler Baustein für die Weiterentwicklung der Kognitionswissenschaft. Die hier präsentierten Ergebnisse integrieren Erkenntnisse aus der Forschung zur Blickbewegungskontrolle und zur Satzverarbeitung beim Lesen und gehen dabei auf Herausforderungen bei der Modellierung von zeitlich geordneten Daten, statistischer Inferenz und interindividueller Variabilität ein. Unter Verwendung von Kerndichteschätzung und einer pseudo-marginalen Wahrscheinlichkeitverteilung für Fixationsdauern und -orte wird eine Implementation für die Likelihood des SWIFT-Modells zur Blickbewegungskontrolle beim Lesen (Engbert et al., Psychological Review, 112, 2005, S. 777–813) eingeführt. Im breiteren Kontext der Datenassimilation wird Bayes'sche Parameterinferenz mit adaptiven Markov-Chain-Monte-Carlo-Techniken verwendet, um eine zuverlässige Modellanpassung zu ermöglichen. In verschiedenen Studien hat sich dieser methodische Rahmen als geeignet erwiesen, um zuverlässige Parameterrückgewinnung aus simulierten Daten und Vorhersage experimenteller Zusammenfassungsstatistiken zu ermöglichen. Trotz dessen Komplexität kann SWIFT innerhalb eines fundierten Bayes'schen Workflows angepasst werden und macht daraufhin zuverlässige Vorhersagen für interindividuelle Unterschiede sowie die Modellierung experimenteller Effekte bei verschiedenen geometrischen Änderungen von Text. Basierend auf diesen Fortschritten wird das integrierte dynamische Modell SEAM eingeführt. Dieses kombiniert die Forschungsgebiete der traditionell psychologisch geprägten Blickbewegungskontrolle und der traditionell psycholinguistisch geprägten postlexikalischen Sprachverarbeitung in Form von cue-basiertem Gedächtnisabruf (Lewis & Vasishth, Cognitive Science, 29, 2005, S. 375–419). Der Nachweis der Durchführbarkeit solcher integrativer Modelle stellt einen bedeutenden Fortschritt bei der natürlichen Sprachverarbeitung beim Lesen dar und legt nahe, dass die vorgestellte Methodik nützlich sein kann, um komplexe kognitive dynamische Modelle zu entwickeln, die Prozesse auf den Ebenen der Wahrnehmung, höheren Kognition, und (okulo-)motorischen Kontrolle integrieren. Diese Erkenntnisse fördern insgesamt die prozessorientierte kognitive Modellierung und betonen die Bedeutung der Bayes'schen Inferenz, individueller Unterschiede und interdisziplinärer Integration für ein ganzheitliches Verständnis von Leseprozessen. Implikationen für Theorie und Methodologie, einschließlich Vorschlägen für Modellvergleich und hierarchische Parameterinferenz, werden kurz diskutiert.
KW  - eye movements
KW  - dynamical cognitive modeling
KW  - sequential likelihood
KW  - psycholinguistics
KW  - cognitive psychology
KW  - sentence processing
KW  - reading
KW  - Blickbewegungen
KW  - Dynamische kognitive Modellierung
KW  - Sequenzielle Likelihood
KW  - Psycholinguistik
KW  - Kognitionspsychologie
KW  - Satzverarbeitung
KW  - Lesen
Y1  - 2024
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-622792
ER  - 
TY  - JOUR
A1  - Malem-Shinitski, Noa
A1  - Opper, Manfred
A1  - Reich, Sebastian
A1  - Schwetlick, Lisa
A1  - Seelig, Stefan A.
A1  - Engbert, Ralf
T1  - A mathematical model of local and global attention in natural scene viewing
JF  - PLoS Computational Biology : a new community journal
N2  - 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.
Y1  - 2020
U6  - https://doi.org/10.1371/journal.pcbi.1007880
SN  - 1553-734X
SN  - 1553-7358
VL  - 16
IS  - 12
PB  - PLoS
CY  - San Fransisco
ER  - 
TY  - THES
A1  - Krejca, Martin Stefan
T1  - Theoretical analyses of univariate estimation-of-distribution algorithms
N2  - 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.
N2  - Optimierung ist ein Hauptbestandteil technologischen Fortschritts und oftmals computergestützt. Da viele Optimierungsprobleme schwer sind, ist es jedoch unrealistisch, eine optimale Lösung in angemessener Zeit zu erwarten. Daher werden Heuristiken verwendet, also Programme, die versuchen hochwertige Lösungen schnell zu erzeugen. Eine konkrete Klasse sind Estimation-of-Distribution-Algorithmen (EDAs), die sich durch das Entwickeln probabilistischer Modelle über dem Problemraum auszeichnen. Ein solches Modell wird genutzt, um neue Lösungen zu erzeugen und damit das Modell zu verfeinern, um im nächsten Schritt mit erhöhter Wahrscheinlichkeit bessere Lösungen zu generieren.

In dieser Arbeit untersuchen wir die Klasse univariater EDAs in der booleschen Domäne, also im Raum aller Bitstrings der Länge n. Das probabilistische Modell eines univariaten EDAs besteht dann aus einem n-dimensionalen Wahrscheinlichkeitsvektor, in dem jede Komponente die Wahrscheinlichkeit angibt, eine 1 an der entsprechenden Stelle zu erzeugen.

Mein Beitrag folgt zwei Hauptrichtungen: Erst untersuchen wir allgemeine inhärente Eigenschaften univariater EDAs. Danach bestimmen wir die erwartete Laufzeit gewisser EDAs auf Benchmarks aus der Theorie. Im ersten Abschnitt charakterisieren wir, wann EDAs unbefangen bezüglich der Problemcodierung sind. Dann untersuchen wir sie in einem Szenario, in dem alle Lösungen gleich gut sind, und zeigen, dass sich ihr Modell schnell zu einem falschen entwickelt, falls es immer so angepasst wird, dass sich im Erwartungswert nichts ändert.

Im zweiten Abschnitt zeigen wir, dass die Algorithmen cGA und MMAS-fp eine verrauschte Variante des klassischen Benchmarks OneMax effizient optimieren, bei der eine Gaussverteilung mit Varianz σ² hinzuaddiert wird. Wir beweisen, dass die Algorithmen das wahre Optimum in polynomieller Zeit bezüglich σ² und n erzeugen. Für den MMAS-fp verallgemeinern wir dieses Ergebnis auf lineare Funktionen. Weiterhin beweisen wir eine Laufzeit von Ω(n log(n)) für den Algorithmus UMDA auf OneMax (ohne Rauschen). Zuletzt führen wir einen neuen Algorithmus ein, der die Benchmarks OneMax und LeadingOnes in O(n log(n)) optimiert, was zuvor für noch keine Heuristik gezeigt wurde.
T2  - Theoretische Analysen univariater Estimation-of-Distribution-Algorithmen
KW  - theory
KW  - estimation-of-distribution algorithms
KW  - univariate
KW  - pseudo-Boolean optimization
KW  - run time analysis
KW  - Theorie
KW  - Estimation-of-Distribution-Algorithmen
KW  - univariat
KW  - pseudoboolesche Optimierung
KW  - Laufzeitanalyse
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-434870
ER  - 
TY  - JOUR
A1  - Hlawenka, Peter
A1  - Siemensmeyer, Konrad
A1  - Weschke, Eugen
A1  - Varykhalov, Andrei
A1  - Sanchez-Barriga, Jaime
A1  - Shitsevalova, Natalya Y.
A1  - Dukhnenko, A. V.
A1  - Filipov, V. B.
A1  - Gabani, Slavomir
A1  - Flachbart, Karol
A1  - Rader, Oliver
A1  - Rienks, Emile D. L.
T1  - Samarium hexaboride is a trivial surface conductor
JF  - Nature Communications
N2  - 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.
Y1  - 2018
U6  - https://doi.org/10.1038/s41467-018-02908-7
SN  - 2041-1723
VL  - 9
PB  - Nature Publ. Group
CY  - London
ER  - 
TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - Sungcharoen, Parada
A1  - Böckmann, Christine
T1  - Convergence rate of the modified Landweber method for solving inverse potential problems
JF  - Mathematics : open access journal
N2  - 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.
KW  - nonlinear operator
KW  - regularization
KW  - modified Landweber method
KW  - discrepancy principle
KW  - logarithmic source condition
Y1  - 2020
U6  - https://doi.org/10.3390/math8040608
SN  - 2227-7390
VL  - 8
IS  - 4
PB  - MDPI
CY  - Basel
ER  - 
TY  - JOUR
A1  - Guggenberger, Tobias
A1  - Chechkin, Aleksei
A1  - Metzler, Ralf
T1  - Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials
JF  - New journal of physics : the open-access journal for physics
N2  - 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.
KW  - diffusion
KW  - Boltzmann distribution
KW  - fractional Brownian motion
Y1  - 2022
U6  - https://doi.org/10.1088/1367-2630/ac7b3c
SN  - 1367-2630
VL  - 24
IS  - 7
PB  - Dt. Physikalische Ges.
CY  - [Bad Honnef]
ER  - 
TY  - JOUR
A1  - Bellingeri, Carlo
A1  - Friz, Peter
A1  - Paycha, Sylvie
A1  - Preiß, Rosa Lili Dora
T1  - Smooth rough paths, their geometry and algebraic renormalization
JF  - Vietnam journal of mathematics
N2  - 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.
KW  - Signatures
KW  - Rough paths
KW  - Cartan's development
KW  - Renormalization
Y1  - 2022
U6  - https://doi.org/10.1007/s10013-022-00570-7
SN  - 2305-221X
SN  - 2305-2228
VL  - 50
IS  - 3
SP  - 719
EP  - 761
PB  - Springer
CY  - Singapore
ER  - 
TY  - JOUR
A1  - Petreska, Irina
A1  - Pejov, Ljupco
A1  - Sandev, Trifce
A1  - Kocarev, Ljupčo
A1  - Metzler, Ralf
T1  - Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting
JF  - Fractal and fractional
N2  - 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.
KW  - rotational diffusion
KW  - memory kernel
KW  - Fokker-Planck equation
KW  - non-exponential relaxation
KW  - autocorrelation function
KW  - complex
KW  - susceptibility
Y1  - 2022
U6  - https://doi.org/10.3390/fractalfract6020088
SN  - 2504-3110
VL  - 6
IS  - 2
PB  - MDPI AG, Fractal Fract Editorial Office
CY  - Basel
ER  - 
TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Moradkhani, H.
A1  - Marshall, L.
A1  - Sharma, Ashish
A1  - Geenens, G.
T1  - Data-driven model uncertainty estimation in hydrologic data assimilation
JF  - Water resources research : WRR / American Geophysical Union
N2  - 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.
KW  - data assimilation
KW  - model error
KW  - uncertainty quantification
KW  - particle filter
KW  - nonparametric statistics
Y1  - 2018
U6  - https://doi.org/10.1002/2018WR022627
SN  - 0043-1397
SN  - 1944-7973
VL  - 54
IS  - 2
SP  - 1252
EP  - 1280
PB  - American Geophysical Union
CY  - Washington
ER  - 
TY  - JOUR
A1  - de Wiljes, Jana
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
T1  - Ensemble transform algorithms for nonlinear smoothing problems
JF  - SIAM journal on scientific computing
N2  - 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.
KW  - data assimilation
KW  - smoother
KW  - localization
KW  - optimal transport
KW  - adaptive
KW  - spread correction
Y1  - 2019
U6  - https://doi.org/10.1137/19M1239544
SN  - 1064-8275
SN  - 1095-7197
VL  - 42
IS  - 1
SP  - A87
EP  - A114
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - McKean-Vlasov SDEs in nonlinear filtering
JF  - SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics
N2  - 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.
KW  - data assimilation
KW  - feedback particle filter
KW  - Poincare inequality
KW  - well-posedness
KW  - nonlinear filtering
KW  - McKean-Vlasov
KW  - mean-field equations
Y1  - 2022
U6  - https://doi.org/10.1137/20M1355197
SN  - 0363-0129
SN  - 1095-7138
VL  - 59
IS  - 6
SP  - 4188
EP  - 4215
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Thapa, Samudrajit
A1  - Park, Seongyu
A1  - Kim, Yeongjin
A1  - Jeon, Jae-Hyung
A1  - Metzler, Ralf
A1  - Lomholt, Michael A.
T1  - Bayesian inference of scaled versus fractional Brownian motion
JF  - Journal of physics : A, mathematical and theoretical
N2  - 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.
KW  - Bayesian inference
KW  - scaled Brownian motion
KW  - single particle tracking
Y1  - 2022
U6  - https://doi.org/10.1088/1751-8121/ac60e7
SN  - 1751-8113
SN  - 1751-8121
VL  - 55
IS  - 19
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Anghileri, Daniela
A1  - Burlando, Paolo
A1  - Sharma, Ashish
A1  - Marshall, Lucy
A1  - Moradkhani, Hamid
T1  - Time-varying parameter models for catchments with land use change
BT  - the importance of model structure
JF  - Hydrology and earth system sciences : HESS
N2  - 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.
Y1  - 2018
U6  - https://doi.org/10.5194/hess-22-2903-2018
SN  - 1027-5606
SN  - 1607-7938
VL  - 22
IS  - 5
SP  - 2903
EP  - 2919
PB  - Copernicus
CY  - Göttingen
ER  - 
TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Anghileri, Daniela
A1  - Burlando, P.
A1  - Sharma, A.
A1  - Marshall, L.
A1  - Moradkhani, H.
T1  - Insights on the impact of systematic model errors on data assimilation performance in changing catchments
JF  - Advances in water resources
N2  - 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.
Y1  - 2017
U6  - https://doi.org/10.1016/j.advwatres.2017.12.006
SN  - 0309-1708
SN  - 1872-9657
VL  - 113
SP  - 202
EP  - 222
PB  - Elsevier
CY  - Oxford
ER  - 
TY  - GEN
A1  - Kolbe, Benedikt Maximilian
A1  - Evans, Myfanwy E.
T1  - Isotopic tiling theory for hyperbolic surfaces
T2  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1347 
KW  - isotopic tiling theory
KW  - Delaney–Dress tiling theory
KW  - mapping class groups
KW  - Orbifolds
KW  - maps on surfaces
KW  - hyperbolic tilings
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-544285
SN  - 1866-8372
IS  - 1
ER  - 
TY  - JOUR
A1  - Leung, Tsz Yan
A1  - Leutbecher, Martin
A1  - Reich, Sebastian
A1  - Shepherd, Theodore G.
T1  - Forecast verification
BT  - relating deterministic and probabilistic metrics
JF  - Quarterly journal of the Royal Meteorological Society
N2  - 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.
KW  - CRPS
KW  - ensembles
KW  - idealised turbulence
KW  - NWP
KW  - RMSE
KW  - verification
Y1  - 2021
U6  - https://doi.org/10.1002/qj.4120
SN  - 0035-9009
SN  - 1477-870X
VL  - 147
IS  - 739
SP  - 3124
EP  - 3134
PB  - Wiley
CY  - Hoboken
ER  - 
TY  - JOUR
A1  - Ayanbayev, Birzhan
A1  - Klebanov, Ilja
A1  - Li, Han Cheng
A1  - Sullivan, Tim J.
T1  - Gamma-convergence of Onsager-Machlup functionals
BT  - I. With applications to maximum a posteriori estimation in Bayesian inverse problems
JF  - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data
N2  - 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.
KW  - Bayesian inverse problems
KW  - Gamma-convergence
KW  - maximum a posteriori
KW  - estimation
KW  - Onsager-Machlup functional
KW  - small ball probabilities;
KW  - transition path theory
Y1  - 2021
U6  - https://doi.org/10.1088/1361-6420/ac3f81
SN  - 0266-5611
SN  - 1361-6420
VL  - 38
IS  - 2
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Lie, Han Cheng
A1  - Stahn, Martin
A1  - Sullivan, Tim J.
T1  - Randomised one-step time integration methods for deterministic operator differential equations
JF  - Calcolo
N2  - 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.
KW  - Time integration
KW  - Operator differential equations
KW  - Randomisation
KW  - Uncertainty quantification
Y1  - 2022
U6  - https://doi.org/10.1007/s10092-022-00457-6
SN  - 0008-0624
SN  - 1126-5434
VL  - 59
IS  - 1
PB  - Springer
CY  - Milano
ER  - 
TY  - JOUR
A1  - Malem-Shinitski, Noa
A1  - Ojeda, Cesar
A1  - Opper, Manfred
T1  - Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects
JF  - Entropy
N2  - 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.
KW  - Bayesian inference
KW  - point process
KW  - Gaussian process
Y1  - 2022
U6  - https://doi.org/10.3390/e24030356
SN  - 1099-4300
VL  - 24
IS  - 3
PB  - MDPI
CY  - Basel
ER  - 
TY  - THES
A1  - Malem-Shinitski, Noa
T1  - Bayesian inference and modeling for point processes with applications from neuronal activity to scene viewing
T1  - Bayessche Inferenz und Modellierung für Punktprozesse mit Anwendungen von neuronaler Aktivität bis Szenenbetrachtung
N2  - 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.
N2  - Punktprozesse sind eine gängige Methode zur Modellierung von Ereignismengen. Von Erdbeben bis zu Social-Media-Posts, von den neuronalen Spikes bis zum Zeitpunkt von Verbrechen, von Aktienkursen bis zur Ausbreitung von Krankheiten - diese Phänomene lassen sich auf das Auftreten von Ereignissen reduzieren, die in Punkten konzentriert sind. Häufig treten diese Ereignisse nacheinander auf und bilden eine Zeitreihe.

Modelle von Punktprozessen können verwendet werden, um unser Verständnis solcher Ereignisse für Klassifizierung und Vorhersage zu vertiefen. Solche Modelle umfassen einen zugrunde liegenden Zufallsprozess, der die Ereignisse erzeugt. In dieser Arbeit wird die Bayes'sche Methodik verwendet, um den zugrunde liegenden generativen Prozess aus den beobachteten Daten abzuleiten. Wir leisten einen doppelten Beitrag: Wir entwickeln neue Modelle und neue Inferenzmethoden für diese Prozesse.

Wir schlagen ein Modell vor, das die Familie der Punktprozesse erweitert, bei denen das Auftreten eines Ereignisses von den vorherigen Ereignissen abhängt. Diese Familie ist als Hawkes-Prozesse bekannt. Während in den meisten bestehenden Modellen solcher Prozesse davon ausgegangen wird, dass vergangene Ereignisse nur eine exzitatorische Wirkung auf zukünftige Ereignisse haben, konzentrieren wir uns auf den neu entwickelten nichtlinearen Hawkes-Prozess, bei dem vergangene Ereignisse exzitatorische und hemmende Wirkungen haben können. Nach der Definition des Modells stellen wir seine Inferenzmethode vor und wenden sie auf Daten aus verschiedenen Bereichen an, unter anderem auf die neuronale Aktivität.

Das zweite Modell, das in dieser Arbeit beschrieben wird, betrifft einen speziellen Fall von Punktprozessen - den Entscheidungsprozess, der der menschlichen Blicksteuerung zugrunde liegt. Dieser Prozess führt zu einer Reihe von fixierten Positionen in einem Bild. Wir haben ein neues Modell entwickelt, um diesen Prozess zu beschreiben, motiviert durch das bekannte Exploration-Exploitation-Dilemma. Neben dem Modell stellen wir einen Bayes'schen Inferenzalgorithmus vor, um die Modellparameter abzuleiten.

Wir bleiben auf dem Gebiet der menschlichen Szenenbetrachtung und stellen fest, dass es in diesem Bereich keine bewährten Verfahren für die Bayes'sche Inferenz gibt. Wir geben einen Überblick über vier gängige Algorithmen und vergleichen ihre Leistungen bei der Ableitung von Parametern für zwei Scanpfadmodelle.

Die in dieser Dissertation vorgestellten neuen Modelle und Inferenzalgorithmen bereichern das Verständnis von Punktprozessdaten und ermöglichen es uns, sinnvolle Erkenntnisse zu gewinnen.
KW  - Bayesian inference
KW  - point process
KW  - statistical machine learning
KW  - sampling
KW  - modeling
KW  - Bayessche Inferenz
KW  - Modellierung
KW  - Punktprozess
KW  - Stichprobenentnahme aus einem statistischen Modell
KW  - statistisches maschinelles Lernen
Y1  - 2023
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-614952
ER  - 
TY  - JOUR
A1  - Ayanbayev, Birzhan
A1  - Klebanov, Ilja
A1  - Lie, Han Cheng
A1  - Sullivan, Tim J.
T1  - Gamma-convergence of Onsager-Machlup functionals
BT  - II. Infinite product measures on Banach spaces
JF  - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data
N2  - 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.
KW  - Bayesian inverse problems
KW  - Gamma-convergence
KW  - maximum a posteriori
KW  - estimation
KW  - Onsager-Machlup functional
KW  - small ball probabilities
KW  - transition path theory
Y1  - 2021
U6  - https://doi.org/10.1088/1361-6420/ac3f82
SN  - 0266-5611
SN  - 1361-6420
VL  - 38
IS  - 2
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - GEN
A1  - Ehlen, Tobias
A1  - Flöge, Annie
A1  - Göbel, Franziska
A1  - Keller, Peter
A1  - Rœlly, Sylvie
ED  - Keller, Peter
ED  - Rœlly, Sylvie
T1  - Übungsbuch zur Stochastik
BT  - Aufgaben und Lösungen  ; Grundlegende Konzepte und Anwendungen
N2  - Dieses Buch stellt Übungen zu den Grundbegriffen und Grundsätzen der Stochastik und ihre Lösungen zur Verfügung. So wie man Tonleitern in der Musik trainiert, so berechnet man Übungsaufgaben in der Mathematik. In diesem Sinne soll dieses Übungsbuch vor allem als Vorlage dienen für das eigenständige, eigenverantwortliche Lernen und Üben.
Die Schönheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Phänomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Glücksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.
N2  - This book provides exercises on the basic concepts and principles of stochastics and their solutions. Just as one trains scales in music, one calculates exercises in mathematics. In this sense, this exercise book is primarily intended to serve as a template for independent learning and practice.
The beauty and uniqueness of probability theory is that it can model a variety of real phenomena. Therefore, one can find exercises with connections to geometry, gambling, actuarial mathematics, demography and many other topics.
KW  - Aufgabensammlung
KW  - Wahrscheinlichkeitstheorie
KW  - Stochastik
KW  - Wahrscheinlichkeitsverteilung
KW  - Zufallsvariable
KW  - Grenzwertsatz
KW  - Konfidenzintervall
KW  - exercise collection
KW  - probability theory
KW  - stochastics
KW  - probability distribution
KW  - random variable
KW  - limit theorem
KW  - confidence interval
Y1  - 2023
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-595939
SN  - 978-3-86956-563-7
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Lazarides, Rebecca
A1  - Rubach, Charlott
T1  - Instructional characteristics in mathematics classrooms
BT  - relationships to achievement goal orientation and student engagement
JF  - Mathematics Education Research Journal
N2  - 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.
KW  - Mathematics classrooms
KW  - Instruction
KW  - Intrinsicmotivation
KW  - Achievement goal orientation
KW  - Effort
KW  - Gender
Y1  - 2017
U6  - https://doi.org/10.1007/s13394-017-0196-4
SN  - 1033-2170
SN  - 2211-050X
VL  - 29
SP  - 201
EP  - 217
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Eshghi, Nasim
A1  - Mach, Thomas
A1  - Reichel, Lothar
T1  - New matrix function approximations and quadrature rules based on the Arnoldi process
JF  - Journal of computational and applied mathematics
N2  - 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.
KW  - Arnoldi process
KW  - Matrix function approximation
KW  - Quadrature rule
Y1  - 2021
U6  - https://doi.org/10.1016/j.cam.2021.113442
SN  - 0377-0427
SN  - 1879-1778
VL  - 391
PB  - Elsevier
CY  - Amsterdam
ER  - 
TY  - BOOK
A1  - Zass, Alexander
A1  - Zagrebnov, Valentin
A1  - Sukiasyan, Hayk
A1  - Melkonyan, Tatev
A1  - Rafler, Mathias
A1  - Poghosyan, Suren
A1  - Zessin, Hans
A1  - Piatnitski, Andrey
A1  - Zhizhina, Elena
A1  - Pechersky, Eugeny
A1  - Pirogov, Sergei
A1  - Yambartsev, Anatoly
A1  - Mazzonetto, Sara
A1  - Lykov, Alexander
A1  - Malyshev, Vadim
A1  - Khachatryan, Linda
A1  - Nahapetian, Boris
A1  - Jursenas, Rytis
A1  - Jansen, Sabine
A1  - Tsagkarogiannis, Dimitrios
A1  - Kuna, Tobias
A1  - Kolesnikov, Leonid
A1  - Hryniv, Ostap
A1  - Wallace, Clare
A1  - Houdebert, Pierre
A1  - Figari, Rodolfo
A1  - Teta, Alessandro
A1  - Boldrighini, Carlo
A1  - Frigio, Sandro
A1  - Maponi, Pierluigi
A1  - Pellegrinotti, Alessandro
A1  - Sinai, Yakov G.
ED  - Roelly, Sylvie
ED  - Rafler, Mathias
ED  - Poghosyan, Suren
T1  - Proceedings of the XI international conference stochastic and analytic methods in mathematical physics
N2  - 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.
T3  - Lectures in pure and applied mathematics - 6 
KW  - statistical mechanics
KW  - random point processes
KW  - stochastic analysis
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-459192
SN  - 978-3-86956-485-2
SN  - 2199-4951
SN  - 2199-496X
IS  - 6
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Omel'chenko, Oleh
A1  - Laing, Carlo R.
T1  - Collective states in a ring network of theta neurons
JF  - Proceedings of the Royal Society of London. Series A, Mathematical, physical and engineering sciences
N2  - 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.
KW  - theta neurons
KW  - neural networks
KW  - bumps
Y1  - 2022
U6  - https://doi.org/10.1098/rspa.2021.0817
SN  - 1364-5021
SN  - 1471-2946
VL  - 478
IS  - 2259
PB  - Royal Society
CY  - London
ER  - 
TY  - JOUR
A1  - Kolbe, Benedikt Maximilian
A1  - Evans, Myfanwy E.
T1  - Enumerating isotopy classes of tilings guided by the symmetry of triply
JF  - Siam journal on applied algebra and geometry
N2  - 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.
KW  - isotopic tiling theory
KW  - mapping class group
KW  - orbifolds
KW  - group
KW  - presentations
KW  - representations of groups as automorphism groups of
KW  - algebraic systems
KW  - triply periodic minimal surface
KW  - Delaney--Dress
KW  - tiling theory
KW  - hyperbolic tilings
KW  - two-dimensional topology
Y1  - 2022
U6  - https://doi.org/10.1137/20M1358943
SN  - 2470-6566
VL  - 6
IS  - 1
SP  - 1
EP  - 40
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - THES
A1  - Wallenta, Daniel
T1  - Sequences of compact curvature
T1  - Sequenzen mit kompakter Krümmung
N2  - 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.
N2  - Die Theorie der Sequenzen mit kompakter Krümmung, sogenannter Quasikomplexe, ist eine Verallgemeinerung der Theorie der Fredholm Komplexe. Um ein Verständnis für (Quasi-)Komplexe zu gewinnen, müssen Inhalte aus verschiedenen Teilgebieten der Mathematik kombiniert werden. Alle hierfür wesentlichen Grundlagen sind im ersten Kapitel dieser Dissertation zusammengefasst. Dies betrifft unter anderem gewisse Elemente der Funktionalanalysis und der Differentialgeometrie, sowie die Theorie der klassischen Pseudodifferentialoperatoren. Im zweiten Kapitel wird anschließend die abstrakte Theorie der Quasikomplexe und zugehöriger Quasimorphismen im Kontext der Funktionalanalysis entwickelt. Dabei werden verschiedene Typen von Quasikomplexen und Quasimorphismen klassifiziert, deren Eigenschaften analysiert und Beispiele betrachtet. Ein zentraler Punkt hierbei ist die Lösung des Problems, für welche dieser Objekte sich eine besondere charakteristische Zahl, die sogenannte Lefschetz-Zahl, definieren lässt. Die dargestellten Resultate zeigen, dass die in dieser Arbeit gegebene Definition eine natürliche Erweiterung der klassischen Lefschetz-Zahl darstellt. Abschließend wird die entwickelte Theorie im dritten Kapitel auf elliptische Quasikomplexe von Pseudodifferentialoperatoren angewendet. Dabei werden insbesondere Verallgemeinerungen der berühmten Atiyah-Singer-Index-Formel und des Lefschetz-Fixpunkt-Theorems von Atiyah and Bott bewiesen.
KW  - Index Theorie
KW  - Fredholm Komplexe
KW  - Elliptische  Komplexe
KW  - Index theory
KW  - Elliptic complexes
KW  - Fredholm complexes
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-87489
ER  - 
TY  - THES
A1  - Wichitsa-nguan, Korakot
T1  - Modifications and extensions of the logistic regression and Cox model
T1  - Modifikationen und Erweiterungen des logistischen Regressionsmodells und des Cox-Modells
N2  - 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.
N2  - In vielen statistischen Anwendungen besteht die Aufgabe darin, die Beziehung zwischen Einflussgrößen und einer Zielgröße zu modellieren. Die Wahl eines geeigneten Modells hängt vom Typ der Zielgröße und vom Ziel der Untersuchung ab - während lineare Modelle für die Beschreibung des Zusammenhanges stetiger Outputs und Einflussgrößen genutzt werden, dienen logistische Regressionsmodelle zur Modellierung binärer Zielgrößen und das Cox-Modell zur Modellierung von Lebendauer-Daten.
In epidemiologischen, medizinischen, biologischen, sozialen und ökonomischen Studien wird oftmals die logistische Regression angewendet, um den Zusammenhang zwischen einer binären Zielgröße und den erklärenden Variablen, den Kovariaten, zu modellieren. In epidemiologischen Studien muss häufig eine große Anzahl von Individuen für eine lange Zeit beobachtet werden. Um hierbei Kosten zu reduzieren, wird ein "Case-Cohort-Design" angewendet. Hierbei werden die Einflussgrößen nur für die Individuen erfasst, für die das interessierende Ereignis eintritt, und für eine zufällig gewählte kleine Teilmenge von Individuen, die Subkohorte.
In der vorliegenden Arbeit wird das Schätzen im logistischen Regressionsmodell unter Case-Cohort-Design betrachtet. Für den Fall, dass auch die Kovariate binär ist, wurde bereits von Prentice (1986) die asymptotische Normalität des Maximum-Likelihood-Schätzers für den Logarithmus des "odds ratio", einen Parameter, der den Effekt der Kovariate charakterisiert, angegeben. In dieser Arbeit wird über einen Maximum-Likelihood-Zugang ein Schätzer für die Varianz der Grenzverteilung hergeleitet, für den durch empirische Untersuchungen gezeigt wird, dass er dem von Prentice überlegen ist.
Ausgehend von dem binärem Kovariate-Modell werden Maximum-Likelihood-Schätzer für logistische Regressionsmodelle mit diskreten Kovariaten unter Case-Cohort-Design hergeleitet. Die asymptotische Normalität wird gezeigt; darauf aufbauend können Formeln für die Standardfehler angegeben werden. Simulationsstudien ergänzen diesen Abschnitt. Sie zeigen den Einfluss des Umfanges der Subkohorte auf die Varianz der Schätzer.
Logistische Regression ist geeignet, wenn man das interessierende Ereignis für alle Individuen beobachten kann und wenn man ein festes Zeitintervall betrachtet. Will man die Zeit bis zum Eintreten eines Ereignisses bei der Untersuchung der Wirkung der Kovariate berücksichtigen, so sind Lebensdauermodelle angemessen. Hierbei können auch zensierte Daten behandelt werden. Ein sehr häufig angewendetes Regressionsmodell ist das von Cox (1972) vorgeschlagene, bei dem die Hazardrate durch
λ(t|x) = λ0(t) exp(β^Tx)
definiert ist. Hierbei ist λ0(t) eine unspezifizierte Baseline-Hazardrate und X ist ein Kovariat-Vektor, β ist ein p-dimensionaler Koeffizientenvektor.
Nachdem ein Überblick über das Schätzen und Testen im Cox-Modell und seinen Erweiterungen gegeben wird, werden Aussagen zur Schätzbarkeit des Parameters β durch die "partiallikelihood"- Methode hergeleitet. Grundlage hierzu sind neue Darstellungen der beobachteten Fisher-Information, die die Ergebnisse von Andersen and Gill (1982) erweitern. Unter Regularitätsbedingungen ist der Schätzer asymptotisch normal; die Inverse der Grenzmatrix der Fisher-Information ist die Varianzmatrix der Grenzverteilung. Bedingungen für die Nichtsingularität dieser Grenzmatrix führen zum Begriff der asymptotischen Schätzbarkeit, der in der vorliegenden Arbeit ausführlich untersucht wird. Darüber hinaus ist diese Matrix Grundlage für die Herleitung lokal optimaler Kovariate. In einer Sensitivitätsanalyse wird die Effizienz gewählter Kovariate berechnet. Die Berechnungen zeigen, dass die Baseline-Verteilung nur wenig Einfluss auf die Effizienz hat. Entscheidend ist die Wahl der Kovariate. Es werden einige Vorschläge für anwendbare optimale Kovariate und Berechnungsverfahren für das Auffinden optimaler Kovariate diskutiert.
Eine Erweiterung des Cox-Modells besteht darin, zeitabhängige Koeffizienten zuzulassen. Da diese Koeffizientenfunktionen nicht näher spezifiziert sind, werden sie nichtparametrisch geschätzt. Eine mögliche Methode ist die "local-linear-partial-likelihood"-Methode, deren Eigenschaften beispielsweise in der Arbeit von Cai and Sun (2003) untersucht wurden. In der vorliegenden Arbeit werden Simulationen zu dieser Methode durchgeführt.
Hauptaspekt ist das Testen der Koeffizientenfunktion. Getestet wird, ob diese Funktion eine bestimmte parametrische Form besitzt. Betrachtet wird der Score-Vektor, der von der "localconstant-partial-likelihood"-Funktion abgeleitet wird. Ausgehend von der asymptotischen Normalität dieses Vektors an verschiedenen Gitterpunkten kann gezeigt werden, dass die Verteilung der geeignet standardisierten quadratischen Form unter der Nullhypothese gegen eine Chi-Quadrat-Verteilung konvergiert.
Die Eigenschaften des auf dieser Grenzverteilungsaussage aufbauenden Tests hängen nicht nur vom Stichprobenumfang, sondern auch vom verwendeten Glättungsparameter ab. Deshalb ist es sinnvoll, auch einen Bootstrap-Test zu betrachten. In der vorliegenden Arbeit wird ein Bootstrap-Test zum Testen der Hypothese, dass die Koeffizienten-Funktion konstant ist, d.h. dass das klassische Cox-Modell vorliegt, vorgeschlagen. Der Algorithmus wird angegeben. Simulationen zum Verhalten dieses Tests unter der Nullhypothese und einer speziellen Alternative werden durchgeführt.

Literatur
P. K. Andersen and R. D. Gill. Cox's regression model for counting processes: a large sample study. Ann. Statist., 10(4):1100{1120, 1982.

Z. Cai and Y. Sun. Local linear estimation for time-dependent coefficients in Cox's regression models. Scand. J. Statist., 30(1):93-111, 2003.

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.
KW  - survival analysis
KW  - Cox model
KW  - logistic regression analysis
KW  - logistische Regression
KW  - Case-Cohort-Design
KW  - Cox-Modell
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-90033
ER  - 
TY  - INPR
A1  - Makhmudov, Olimdjan
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - The first mixed problem for the nonstationary Lamé system
N2  - We find an adequate interpretation of the Lamé operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lamé system.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)10 
KW  - Lamé system
KW  - evolution equation
KW  - first boundary value problem
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71923
SN  - 2193-6943
VL  - 3
IS  - 10
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Ziese, Ramona
T1  - Geometric electroelasticity
T1  - Geometrische Elektroelastizität
N2  - 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.
N2  - In der vorliegenden Arbeit wird eine diffentialgeometrische Formulierung der Elektroelastizitätstheorie entwickelt, die auch thermische und magnetische Einflüsse berücksichtigt. Hierbei wird die Bewegung von Körpern untersucht, die aus einem elastischen Material bestehen und sich durch mechanische Kräfte, Wärmezufuhr und den Einfluss eines äußeren elektromagnetischen Feldes verformen. Dazu werden physikalische Bilanzgleichungen (Massenerhaltung, Impuls-, Drehimpuls- und Energiebilanz) aufgestellt, um mit deren Hilfe eine Gleichung zu formulieren, die die Bewegung des Körpers während der Deformation beschreibt. Dabei werden sowohl der Körper als auch der umgebende Raum als Riemannsche Mannigfaltigkeiten modelliert, wobei zugelassen ist, dass der Körper eine geringere Dimension hat als der ihn umgebende Raum. Auf diese Weise kann man nicht nur - wie sonst üblich - die Deformation dreidimensionaler Körper im dreidimensionalen euklidischen Raum beschreiben, sondern auch die Deformation von Membranen und die Deformation innerhalb eines gekrümmten Raums. Weiterhin werden sogenannte konstitutive Gleichungen formuliert, die die Eigenschaften des verwendeten Materials kodieren. Die Energiebilanz ist eine skalare Gleichung und kann daher leicht auf Riemannschen Mannigfaltigkeiten formuliert werden. Es wird gezeigt, dass die Forderung der Invarianz der Energiebilanz unter der Wirkung von beliebigen Diffeomorphismen auf den umgebenden Raum bereits die restlichen Bilanzgleichungen impliziert. Das verallgemeinert ein Resultat von Marsden und Hughes, das nur für Körper anwendbar ist, die die selbe Dimension wie der umgebende Raum haben und keine elektromagnetischen Felder berücksichtigt. Üblicherweise wird in Arbeiten über Elektroelastizität die Entropieungleichung verwendet, um zu entscheiden, welche Deformationen physikalisch zulässig sind und welche nicht. Sie wird außerdem verwendet, um Einschränkungen für die möglichen Formen von konstitutiven Gleichungen, die das Material beschreiben, herzuleiten. Leider gehen die Meinungen über die physikalisch korrekte Formulierung der Entropieungleichung auseinander sobald elektromagnetische Felder beteiligt sind. Weiterhin ist unklar, wie die Entropieungleichung für den Fall einer Membran, die einem elektromagnetischen Feld ausgesetzt ist, formuliert werden muss. Daher zeigen wir, dass die Benutzung der Entropieungleichung ersetzt werden kann durch die Forderung, dass für einen gegebenen Prozess die Energiebilanz invariant ist unter der Wirkung eines beliebigen Diffeomorphimus' auf den umgebenden Raum und der linearen Reskalierung der Temperatur. Zum einen liefert diese Forderung die gewünschten Einschränkungen für die Form der konstitutiven Gleichungen, zum anderen benoetigt sie viel schwächere Annahmen als die übliche Argumentation mit der Entropieungleichung, die man in der Physikliteratur findet. Unser Resultat ist dabei wieder eine Verallgemeinerung eines Theorems von Marsden und Hughes, wobei es, so wie deren Resultat, nur für Körper gilt, die als offene Teilmengen des dreidimensionalen euklidischen Raums modelliert werden können.
KW  - Elastizität
KW  - Elektrodynamik
KW  - Mannigfaltigkeit
KW  - konstitutive Gleichungen
KW  - Bewegungsgleichung
KW  - elasticity
KW  - electrodynamics
KW  - manifold
KW  - constitutive relations
KW  - equation of motion
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72504
ER  - 
TY  - THES
A1  - Vu, Dinh Phuong
T1  - Using video study to investigate eighth-grade mathematics classrooms in Vietnam
T1  - Die Nutzung von Videostudien zur Untersuchung des Mathematikunterrichts in der 8. Klasse in Vietnam
N2  - 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.
N2  - Das International Project for the Evaluation of Educational Achievement (IEA) wurde in den 1950er Jahren gegründet. Seitdem führte das IEA viele Studien in Bereich mathematischer Bildung durch, insbesondere die First International Mathematics Study (FIMS) im Jahre 1964, die Second International Mathematics Study (SIMS) in den Jahren 1980–1982 und eine Reihe von Studien, die mit der Third International Mathematics and Science Study (TIMSS) begann und seit 1995 alle vier Jahre durchgeführt wird. Nach Stigler et al. (1999) erreichten US-amerikanische Studenten bei FIMS und SIMS niedrigere Ergebnisse als Schüler anderer Länder (S. 1). Daher wurde TIMSS 1995 erweitert um eine ‘Videotape Classroom Study’ mit dem Ziel, „mehr über die unterrichtlichen und kulturellen Prozesse, die mit Leistung zusammenhängen“, zu erfahren (S. 1; Übersetzung vom engl. Original). Von den Ergebnissen der TIMMS 1995 Video Study ausgehend verglichen Stigler und Hiebert (1999) Unterricht mit „Gebirgszügen, die die Wasseroberfläche durchstoßen“, womit sie ausdrücken sollten, was die Bergspitzen sichtbar, große Teile des Gebirges aber unter dem Wasser verborgen sind (S. 73–78; Übersetzung vom engl. Original). Durch die wiederholte Analyse videographierter Unterrichtsstunden aus Deutschland, Japan und den USA entdeckten sie, dass „die Arten des Unterrichts innerhalb jedes Landes von Stunde zu Stunde ähnlich sind. Zumindest gibt es bestimmte wiederkehrende Aspekte [oder Skripte], welche für viele Stunden eines Landes typisch sind und die Stunden gegenüber anderen Ländern abgrenzen“ (S. 77f.). Sie entdeckten außerdem, dass Unterricht eine „kulturelle Aktivität“ ist, Unterrichtsarten also „verstanden werden müssen in Relation zu den kulturellen Überzeugungen und Annahmen, die sie umgeben“ (S. 85, 88). Hierauf aufbauend war es ein Ziel der Dissertation, kulturelle Aspekte des Mathematikunterricht zu untersuchen und die Ergebnisse mit Mathematikunterricht in Vietnam zu vergleichen. Ein weiteres Ziel war die Erhebung der Charakteristika vietnamesischen Mathematikunterricht durch eine Videostudie in Vietnam und der anschließende Vergleich dieser Charakteristika mit denen anderer Länder. Im Einzelnen befasste sich diese Dissertation mit den folgenden Forschungszielen: - Untersuchung der Charakteristika von Lehren und Lernen in unterschiedlichen Kulturen und vorläufiger Vergleich der Resultate mit dem Lehren und Lernen von Mathematik in Vietnam - Einführung der TIMSS und der TIMSS Video Study und der methodologischen Vorteile von Videostudien für die Untersuchung von Mathematikunterricht in Vietnam - Durchführung der Videostudie in Vietnam, um Unterrichtsskripte des Mathematikunterrichts in 8. Klassen in Vietnam zu identifizieren - Vergleich ausgewählter Aspekte des Mathematikunterrichts in Vietnam mit denen anderer Länder auf der Grundlage der Videostudie in Vietnam und Diskussion von Ähnlichkeiten und Unterschieden zwischen Ländern - Untersuchung der Herausforderungen für eine Innovation der Unterrichtsmethoden im Mathematikunterricht Vietnams Diese Dissertation entstand in der Hoffnung, dass sie eine nützliche Referenz für Lehramtsstudenten zum Verständnis der Natur des Unterrichts und zur Entwicklung der eigenen Lehrerpersönlichkeit darstellen möge.
KW  - Videostudie
KW  - Mathematikunterricht
KW  - Unterrichtsmethode
KW  - TIMSS
KW  - Kulturelle Aktivität
KW  - video study
KW  - mathematics education
KW  - teaching methods
KW  - TIMSS
KW  - Vietnam
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72464
ER  - 
TY  - INPR
A1  - Bagdonavičius, Vilijandas B.
A1  - Levuliene, Ruta
A1  - Nikulin, Mikhail S.
A1  - Zdorova-Cheminade, Olga
T1  - Tests for homogeneity of survival distributions against non-location alternatives and analysis of the gastric cancer data
N2  - 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.
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 03 
KW  - Censoring
KW  - Cross-effects
KW  - Kolmogorov-Smirnov type tests
KW  - Logrank test
KW  - Non-proportional hazards
KW  - Proportional hazards
KW  - Two-sample tests
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51527
ER  - 
TY  - INPR
A1  - Gairing, Jan
A1  - Högele, Michael
A1  - Kosenkova, Tetiana
A1  - Kulik, Alexei Michajlovič
T1  - On the calibration of Lévy driven time series with coupling distances : an application in paleoclimate
N2  - 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 Lévy 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 Lévy 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 Lévy component for some tail index greater than 2.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 2 
KW  - time series with heavy tails
KW  - index of stability
KW  - goodness-of-fit
KW  - empirical Wasserstein distance
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69781
SN  - 2193-6943
VL  - 3
IS  - 2
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Aizenberg, Lev A.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - An integral formula for the number of lattice points in a domain
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 3 
KW  - logarithmic residue
KW  - lattice point
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70453
SN  - 2193-6943
VL  - 3
IS  - 3
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Wallenta, Daniel
T1  - A Lefschetz fixed point formula for elliptic quasicomplexes
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)12 
KW  - Perturbed complexes
KW  - curvature
KW  - Lefschetz number
KW  - fixed point formula
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-67016
ER  - 
TY  - INPR
A1  - Högele, Michael
A1  - Pavlyukevich, Ilya
T1  - Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed Lévy type noise
N2  - 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 Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 5 
KW  - hyperbolic dynamical system
KW  - Morse-Smale property
KW  - stable limit cycle
KW  - small noise asymptotic
KW  - multiplicative noise
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70639
SN  - 2193-6943
VL  - 3
IS  - 5
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Conforti, Giovanni
A1  - Dai Pra, Paolo
A1  - Roelly, Sylvie
T1  - Reciprocal class of jump processes
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 6 
KW  - reciprocal processes
KW  - stochastic bridges
KW  - jump processes
KW  - compound Poisson processes
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70776
SN  - 2193-6943
VL  - 3
IS  - 6
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Hübner, Andrea
T1  - Ein multityper Verzweigungsprozess als Modell zur Untersuchung der Ausbreitung von Covid-19
T1  - Modeling the spread of Covid-19 using a multitype branching process
N2  - Im Zuge der Covid-19 Pandemie werden zwei Werte täglich 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. Für die Schätzung dieses Wertes gibt es viele Möglichkeiten - auch das Robert Koch-Institut gibt in seinem täglichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere Möglichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu schätzen. 

In der ersten Hälfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man für die Modellierung benötigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverständnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingeführt und die Ergebnisse aus diesem Themengebiet, die für diese Arbeit wichtig sind, präsentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschränken wir uns auf zwei Typen. Das Prinzip lässt 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 über einen längeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R.  

In der zweiten Hälfte dieser Arbeit stellen wir eine Anwendung der Theorie über Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Veröffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' über einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Schätzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die Möglichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsfälle) zu schätzen. Wir wenden die Schätzer auf die Zahlen von Deutschland an und werten diese schließlich aus.
N2  - During the Covid-19 pandemic, the discussion about the situation has been dominated by two numbers: the number of daily new infected individuals and the reproduction rate. The latter is the average number of people, one infected individual will infect with the disease. Because the number of registered infected individuals is generally not equal to the actual number of people who carry the Corona virus, many facts about the pandemic have to be estimated and can not be known for certain. Since the reproduction rate is an important parameter to signify the course of the Pandemic, many ways to estimate it have been developed. The Institute of Robert Koch in Germany uses two reproduction rates R in their daily reports: The 4-days-R-value and the less fluctuating 7-days-Rvalue. This master thesis will develop another model to estimate the R-value and other interesting aspects of the pandemic.

The first part of this thesis is dedicated to the mathematical foundations needed to understand the model. The reader is expected to already have basic understanding of stochastic processes. In the section Grundlagen we will discuss branching processes and present the results of their theory that are important for our work. We start by introducing simple branching processes and expand the results to multitype branching processes. In service of a simpler notation we will only consider twotype branching processes, but the results can be used for any number of types.

The importance of the parameter λ shall be stressed. It can be seen as the average number of descendants of one individual and dictates the dynamic of the process over a long period of time. Applied to the modeling of the pandemic, λ plays the same role as the reproduction rate R.

In the second part of this thesis will present an application of the previously developed theory about multitype branching processes. Prof. Yanev and his colleagues modeled in their publication Branching stochastic processes as models of Covid-19 epidemic development the spreading of the Corona virus by using a branching process with two types. We will discuss this model and deduce estimators from it. We want to estimate the reproduction rate and find a way to determine the number of not registered infected individuals. The estimators will be applied to the data from Germany and we will discuss the results.
KW  - Covid-19
KW  - Corona
KW  - Reproduktionsrate
KW  - Verzweigungsprozess
KW  - Modellierung
KW  - Covid-19
KW  - corona virus
KW  - reproduction rate
KW  - branching process
KW  - modeling
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-509225
ER  - 
TY  - INPR
A1  - Dyachenko, Evgueniya
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Singular perturbations of elliptic operators
N2  - 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'.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 1 
KW  - singular perturbation
KW  - pseudodifferential operator
KW  - ellipticity with parameter
KW  - regularization
KW  - asymptotics
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69502
SN  - 2193-6943
VL  - 3
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Samaras, Stefanos
T1  - Microphysical retrieval of non-spherical aerosol particles using regularized inversion of multi-wavelength lidar data
T1  - Retrieval der Mikrophysik von nichtkugelförmigen Aerosolpartikeln durch regularisierte Inversion von Mehrwellenlängen-Lidardaten
N2  - 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.
N2  - Zahlreiche Berichte von relativ schnellen Klimaveränderungen im vergangenen Jahrhundert liefern überzeugende Argumente über die Auswirkungen von Aerosolen und Wolken auf Wetter und Klima. Aerosole und Wolken wurden als Quellen größter Unsicherheit in Klimaprognosen identifiziert. Die Strahlungsbilanz der Erde wird verändert durch die Partikelgröße, ihre Morphologie und ihre chemische Zusammensetzung. Konkurrierende Effekte in der Atmosphäre können durch die Bestimmung von mikrophysikalischen Partikeleigenschaften weiter untersucht werden, was der Fokus der vorliegenden Arbeit ist.
Die Aerosolgrößenverteilung, der Brechungsindex der Partikeln und die Einzel-Streu-Albedo sind solche häufig verwendeten Parameter, die mit dem Aerosoltyp und dem Strahlungsantrieb verbunden sind. Hoch entwickelte Lidare (Light Detection and Ranging) haben die Aerosolüberwachung und die optische Profilierung zu einem Routineprozess gemacht. Lidar-Daten wurden verwendet um die Größenverteilung zu bestimmen, was durch die Inversion des sogenannten Lorenz-Mie-Modells (LMM) gelingt. Dieses Modell bietet eine angemessene Behandlung für sphärisch angenäherte Partikeln, es stellt aber keine brauchbare Beschreibung für andere natürlich auftretende beliebig geformte Partikeln -wie z.B. Staubpartikeln- bereit. Andererseits stellt die Einbeziehung einer nicht kugelförmigen Geometrie –wie z.B. einfache Sphäroide- bestimmte optische Eigenschaften mit verbesserter Genauigkeit dar. Angesichts dieser Tatsache erweitern wir das LMM durch die Approximation von Sphäroid-Partikeln. Dazu ist es notwendig den Begriff einer zweidimensionalen Größenverteilung einzuführen. 
Die Inversion einer sehr geringen Anzahl optischer Datenpunkte zur Bestimmung der Form der Größenverteilung ist als ein nichtlineares schlecht gestelltes Problem bekannt. Eine kurze mathematische Analyse wird vorgestellt, die die inhärente Tendenz zu stark oszillierenden Lösungen zeigt. Weiterhin werden Optionen für eine verallgemeinerte Lösung durch Regularisierungsmethoden untersucht und der Grad der Schlechtgestelltheit quantifiziert. Letzteres wird unser Verständnis für die Hauptursache der Instabilität bei den berechneten Lösungsräumen verbessern. Der neue Ansatz ermöglicht es uns, zusätzliche Lidar-Datenpunkte aus Depolarisationsmessungen zu nutzen, die sich aus der Nicht-sphärizität der Partikeln assoziieren. Die Verallgemeinerung des LMMs erhöht erheblich die Komplexität des Problems. Die zugrundeliegende Theorie für die Berechnung der beteiligten optischen Querschnitte (T-Matrix-Ansatz) ist rechnerisch so aufwendig, dass eine Neuberechnung dieser nicht sinnvoll erscheint. Darüber hinaus wird ein zweidimensionales Kollokationsverfahren für die Diskretisierung der Modellgleichung vorgeschlagen. Dieses Verfahren beinhaltet Doppelintegrationen, die wiederum zeitaufwendig sind. Wir überwinden diese Schwierigkeiten durch Verwendung vorgerechneter Datenbanken sowie einer hochentwickelten Retrieval-Software (SphInX: Spheroidal Inversion eXperiments). Diese Software wurde speziell für unseren Zweck entwickelt und ist in der Lage mehrere Datensatzinversionen gleichzeitig durchzuführen und eine große Auswahl von mikrophysikalischen Retrieval-Ausgaben bereitzustellen. 
Eine hybride Regularisierung in Verbindung mit einem Minimierungsverfahren wird als Grundlage für unsere Algorithmen verwendet. Synthetische Daten-Inversionen werden mit verschiedenen atmosphärischen Szenarien durchgeführt, um die Effizienz verschiedener Regularisierungsmethoden zu untersuchen. Die Lücke in der gegenwärtigen wissenschaftlichen Literatur gewisse Unsicherheiten durch breitgefächerte numerische Fälle bereitzustellen, ist ein Hauptanliegen dieser Arbeit. Motiviert davon werden die am besten geeigneten Verfahren einer gründlichen Analyse in Bezug auf ihr Gesamtverhalten, d.h. Genauigkeit und Stabilität, unterzogen. Der allgemeine Trend der Anfangsgrößenverteilung wird in unseren numerischen Experimenten erfasst. Zusätzlich hängt die Rekonstruktionsqualität vom Datenfehler ab. Darüber hinaus wird die Anzahl der notwendigen Depolarisationspunkte zum ersten Mal aus der Sicht des mikrophysikalischen Parameter-Retrievals erforscht. Abschließend verwenden wir unsere Software für verschiedene Messfälle, was weitere Einblicke für künftige Verbesserungen des Algorithmus gibt.
KW  - microphysics
KW  - retrieval
KW  - lidar
KW  - aerosols
KW  - regularization
KW  - ill-posed
KW  - inversion
KW  - Mikrophysik
KW  - Retrieval
KW  - Lidar
KW  - Aerosole
KW  - Regularisierung
KW  - schlecht gestellt
KW  - Inversion
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-396528
ER  - 
TY  - THES
A1  - Santilli, Mario
T1  - Higher order rectifiability in Euclidean space
T1  - Rektifizierbarkeit höherer Ordnung in Euklidischen Räumen
N2  - 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.
N2  - Das erste Ziel dieser Arbeit ist die Entwicklung eines Konzepts zur Beschreibung von Differenzierbarkeit höherer Ordnung für Teilmengen des euklidischen Raumes, welche es erlaubt von höherer Ordnung rektifizierbare Mengen zu charakterisieren. Wir betonen, dass wir für jede Teilmenge A des euklidischen Raumes und jede ganze Zahl k ≥ 2 ein approximatives Differenzial der Ordnung k einführen und beweisen, dass es sich dabei um eine Borelfunktion handelt deren Definitionsbereich eine (möglicherweise leere) Borelmenge ist. Unser Konzept könnte hilfreich für die Behandlung von höherer Ordnung rektifizierbarer Mengen in Anwendungen sein.

Das andere Ziel ist die Verallgemeinerung auf beliebige abgeschlossene Mengen eines bekannten Satzes von Alberti über Rektifizierbarkeit zweiter Ordnung des Randes konvexer Körper. Für den Rand eines solchen konvexen Körper liefert Albertis Resultat eine Stratifikation durch von zweiter Ordnung rektifizierbare Teilmengen des Randes basierend auf der Dimension des (konvexen) Normalenkegels. Für eine geeignete Verallgemeinerung dieses Normalenkegels auf allgemeine abgeschlossene Teilmengen des euklidischen Raumes und unter Verwendung einiger Resultate aus dem ersten Teil können wir zeigen dass eine solche Stratifiaktion für alle abgeschlossenen Mengen existiert.
KW  - higher order rectifiability
KW  - approximate differentiability
KW  - Borel functions
KW  - normal bundle
KW  - coarea formula
KW  - Rektifizierbarkeit höherer Ordnung
KW  - approximative Differenzierbarkeit
KW  - Borel Funktionen
KW  - Coarea Formel
KW  - Normalenbündel
Y1  - 2018
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-403632
ER  - 
TY  - INPR
A1  - Bagderina, Yulia Yu.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Differential invariants of a class of Lagrangian systems with two degrees of freedom
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 2 
KW  - equivalence
KW  - invariant
KW  - Euler-Lagrange equations
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63129
ER  - 
TY  - INPR
A1  - Blanchard, Gilles
A1  - Delattre, Sylvain
A1  - Roquain, Étienne
T1  - Testing over a continuum of null hypotheses
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 1 
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56877
ER  - 
TY  - INPR
A1  - Grudsky, Serguey
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)10 
KW  - singular integral equations
KW  - nonsmooth curves
KW  - boundary value problems
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57745
ER  - 
TY  - INPR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Tunneling for a class of difference operators
N2  - 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 Schrödinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 5 
KW  - semi-classical difference operator
KW  - tunneling
KW  - interaction matrix
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56989
ER  - 
TY  - INPR
A1  - Keller, Peter
A1  - Roelly, Sylvie
A1  - Valleriani, Angelo
T1  - On time duality for quasi-birth-and-death processes
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 4 
KW  - continuous time Markov chain
KW  - hitting times
KW  - time duality
KW  - absorbing boundary
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56973
ER  - 
TY  - INPR
A1  - Tarkhanov, Nikolai Nikolaevich
A1  - Wallenta, Daniel
T1  - The Lefschetz number of sequences of trace class curvature
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 3 
KW  - Perturbed complexes
KW  - curvature
KW  - Lefschetz number
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56969
ER  - 
TY  - INPR
A1  - Koppitz, Jörg
A1  - Musunthia, Tiwadee
T1  - Maximal subsemigroups containing a particular semigroup
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 8 
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57465
ER  - 
TY  - THES
A1  - Murr, Rüdiger
T1  - Reciprocal classes of Markov processes : an approach with duality formulae
T1  - Reziproke Klassen von Markov Prozessen : ein Ansatz mit Dualitätsformeln
N2  - 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 Lévy 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.
N2  - Diese Arbeit befasst sich mit der Charakterisierung von Klassen stochastischer Prozesse durch Dualitätsformeln. Es wird insbesondere der in der Literatur bisher unbehandelte Fall reziproker Klassen stochastischer Prozesse mit Sprungen untersucht. Im ersten Teil stellen wir eine neue Formulierung einer Charakterisierung von Prozessen mit unabhängigen Zuwächsen vor. Diese basiert auf der aus dem Malliavinkalkül bekannten Dualitätsformel für Prozesse mit unendlich oft teilbaren Zuwächsen. Wir präsentieren zusätzlich zwei neue Beweismethoden dieser Dualitätsformel, die nicht auf der Chaoszerlegung des Raumes quadratintegrabler Funktionale beruhen. Eine dieser Methoden basiert auf einer partiellen Integrationsformel fur unendlich oft teilbare Zufallsvektoren. In diesem Rahmen ist unsere Charakterisierung eine Verallgemeinerung des Lemma fur Gaußsche Zufallsvariablen von Stein und des Lemma fur Zufallsvariablen mit Poissonverteilung von Chen. Die Allgemeinheit dieser Methode erlaubt uns durch einen ähnlichen Zugang die Charakterisierung unendlich oft teilbarer Zufallsmaße. Im zweiten Teil der Arbeit konzentrieren wir uns auf die Charakterisierung reziproker Klassen ausgewählter Markovprozesse durch Dualitätsformeln. Wir beginnen mit einer Zusammenfassung bereits existierender Ergebnisse zu den reziproken Klassen Brownscher Bewegungen mit Drift. Es ist uns möglich die Charakterisierung solcher reziproken Klassen durch eine Dualitätsformel physikalisch umzudeuten in eine Newtonsche Gleichung. Damit gelingt uns ein Brückenschlag zwischen derartigen Charakterisierungsergebnissen und der Theorie stochastischer Mechanik durch den Interpretationsansatz, sowie der Theorie stochastischer optimaler Steuerung durch den mathematischen Ansatz. Unter Verwendung der Charakterisierung reziproker Klassen durch Dualitätsformeln beweisen wir weiterhin eine Invarianzeigenschaft der reziproken Klasse Browscher Bewegungen mit Drift unter Zeitumkehrung. Es gelingt uns weiterhin neue Resultate im Rahmen reiner Sprungprozesse zu beweisen. Wir beschreiben reziproke Klassen Markovscher Zählprozesse, d.h. Sprungprozesse mit Sprunghöhe eins, und erhalten eine Charakterisierung der reziproken Klasse vermöge einer Dualitätsformel. Diese beinhaltet als Schlüsselterme eine stochastische Ableitung nach den Sprungzeiten, ein kompensiertes stochastisches Integral und eine Invariante der reziproken Klasse. Wir präsentieren außerdem eine Interpretation der Charakterisierung einer reziproken Klasse im Rahmen der stochastischen Steuerungstheorie. Als weitere Anwendung beweisen wir eine Invarianzeigenschaft der reziproken Klasse Markovscher Zählprozesse unter Zeitumkehrung. Einige dieser Ergebnisse werden fur reine Sprungprozesse mit unterschiedlichen Sprunghöhen verallgemeinert. Insbesondere zeigen wir, dass die reziproken Klassen Markovscher Sprungprozesse vermöge reziproker Invarianten unterschieden werden können. Eine Charakterisierung der reziproken Klasse zusammengesetzter Poissonprozesse durch eine Dualitätsformel gelingt unter der Annahme inkommensurabler Sprunghöhen.
KW  - unendliche Teilbarkeit
KW  - Dualitätsformeln
KW  - reziproke Klassen
KW  - Zählprozesse
KW  - stochastische Mechanik
KW  - infinite divisibility
KW  - duality formulae
KW  - reciprocal class
KW  - counting process
KW  - stochastic mechanics
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-62091
ER  - 
TY  - INPR
A1  - Murr, Rüdiger
T1  - Reciprocal classes of Markov processes : an approach with duality formulae
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)26 
KW  - Duality formula
KW  - reciprocal class
KW  - Levy process
KW  - infinite divisibility
KW  - counting process
KW  - Malliavin calculus
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63018
ER  - 
TY  - BOOK
A1  - Zhuchok, Anatolii V.
T1  - Relatively free doppelsemigroups
N2  - 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.
N2  - Eine Doppelalgebra ist eine auf einem Vektorraum definierte Algebra mit zwei binären linearen assoziativen Operationen. Doppelalgebren spielen eine herausragende Rolle in der algebraischen K-Theorie. Wir betrachten Doppelhalbgruppen, d.h Mengen mit zwei binären assoziativen Operationen, welche die Axiome der Doppelhalbgruppe erfüllen. Doppelhalbgruppen sind Veralgemeinerungen von Halbgruppen und sie stehen in Beziehung zu solchen algebraischen Strukturen wie interassoziative Halbgruppen, restriktive Bihalbgruppen, Dimonoiden und Trioden.
In dieser Lecture Notes werden eine Vielzahl von Beispielen für Doppelhalbgruppen und strong Doppelhalbgruppen gegeben. Die Unabhängigkeit der Axiome für Doppelhalbgruppen wird nachgewiesen. Ein freies Produkt in der Varietät der Doppelhalbgruppen wird vorgestellt. Wir konstruieren auch eine freie (kommutative) strong Doppelhalbgruppe, eine freie n-dinilpotent (strong) Doppelhalbgruppe und eine freie Links n-dinilpotent Doppelhalbgruppe. Darüber hinaus werden die kleinste n-nilpotente Kogruenz, die kleinste n-dinilpotente Kongruenz auf der freien (strong) Doppelhalbgruppe und die kleinste n-dinilpotente Kongruenz auf einer freien Doppelhalbgruppe charakterisiert.
Das Buch richtet sich an Graduierte, Doktoranden, Forscher in Algebra und interessierte Leser.
T3  - Lectures in pure and applied mathematics - 5 
KW  - doppelsemigroup
KW  - interassociativity
KW  - free algebra
KW  - semigroup
KW  - congruence
Y1  - 2018
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-407719
SN  - 978-3-86956-427-2
SN  - 2199-4951
SN  - 2199-496X
IS  - 5
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Antoniouk, Alexandra Viktorivna
A1  - Kiselev, Oleg
A1  - Stepanenko, Vitaly
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)25 
KW  - Heat equation
KW  - the first boundary value problem
KW  - characteristic boundary point
KW  - cusp
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61987
ER  - 
TY  - INPR
A1  - Högele, Michael
A1  - Ruffino, Paulo
T1  - Averaging along Lévy diffusions in foliated spaces
N2  - We consider an SDE driven by a Lévy 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 Lévy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to Lévy driven SDEs of Marcus type.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)10 
KW  - Averaging principle
KW  - foliated diffusion
KW  - Lévy diffusions on manifolds
KW  - canonical Marcus integration
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64926
SN  - 2193-6943
ER  - 
TY  - INPR
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)11 
KW  - Sturm-Liouville problems
KW  - discontinuous Robin condition
KW  - root functions
KW  - Lipschitz domains
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57759
SN  - 2193-6943
ER  - 
TY  - INPR
A1  - Nehring, Benjamin
T1  - Construction of point processes for classical and quantum gases
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)14 
KW  - Gibbs point processes
KW  - permanental-
KW  - determinantal point processes
KW  - cluster expansion
KW  - Lévy measure
KW  - infinitely divisible point processes
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59648
ER  - 
TY  - INPR
A1  - Bär, Christian
A1  - Ballmann, Werner
T1  - Boundary value problems for elliptic differential operators of first order
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)18 
KW  - Elliptic operators
KW  - elliptic boundary conditions
KW  - completeness
KW  - coercivity
KW  - boundary regularity
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60023
ER  -