510 Mathematik
Refine
Year of publication
Document Type
- Preprint (373)
- Article (264)
- Doctoral Thesis (76)
- Postprint (45)
- Monograph/Edited Volume (13)
- Other (10)
- Master's Thesis (6)
- Part of a Book (5)
- Conference Proceeding (5)
- Review (3)
Language
- English (754)
- German (46)
- French (3)
- Multiple languages (1)
Keywords
- random point processes (18)
- statistical mechanics (18)
- stochastic analysis (18)
- index (14)
- boundary value problems (12)
- Fredholm property (10)
- regularization (10)
- cluster expansion (9)
- elliptic operators (9)
- data assimilation (8)
Institute
- Institut für Mathematik (740)
- Extern (14)
- Institut für Physik und Astronomie (14)
- Mathematisch-Naturwissenschaftliche Fakultät (14)
- Hasso-Plattner-Institut für Digital Engineering gGmbH (7)
- Institut für Biochemie und Biologie (6)
- Institut für Informatik und Computational Science (5)
- Department Psychologie (4)
- Department Grundschulpädagogik (3)
- Hasso-Plattner-Institut für Digital Engineering GmbH (3)
The estimation of a log-concave density on R is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.
In 1914 Bohr proved that there is an r ∈ (0, 1) such that if a power series converges in the unit disk and its sum has modulus less than 1 then, for |z| < r, the sum of absolute values of its terms is again less than 1. Recently analogous results were obtained for functions of several variables. The aim of this paper is to comprehend the theorem of Bohr in the context of solutions to second order elliptic equations meeting the maximum principle.
The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.
We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.
Although the general development of mathematical abilities in primary school has been the focus of many researchers, the development of place value understanding has rarely been investigated to date. This is possibly due to the lack of conceptual approaches and empirical studies related to this topic. To fill this gap, a theory-driven and empirically validated model was developed that describes five sequential conceptual levels of place value understanding. The level sequence model gives us the ability to estimate general abilities and difficulties in primary school pupils in the development of a conceptual place value understanding. The level sequence model was tried and tested in Germany, and given that number words are very differently constructed in German and in the languages used in South African classrooms, this study aims to investigate whether this level sequence model can be transferred to South Africa. The findings based on the responses of 198 Grade 2-4 learners show that the English translation of the test items results in the same item level allocation as the original German test items, especially for the three basic levels. Educational implications are provided, in particular concrete suggestions on how place value might be taught according to the model and how to collect specific empirical data related to place value understanding.
The past three decades of policy process studies have seen the emergence of a clear intellectual lineage with regard to complexity. Implicitly or explicitly, scholars have employed complexity theory to examine the intricate dynamics of collective action in political contexts. However, the methodological counterparts to complexity theory, such as computational methods, are rarely used and, even if they are, they are often detached from established policy process theory. Building on a critical review of the application of complexity theory to policy process studies, we present and implement a baseline model of policy processes using the logic of coevolving networks. Our model suggests that an actor's influence depends on their environment and on exogenous events facilitating dialogue and consensus-building. Our results validate previous opinion dynamics models and generate novel patterns. Our discussion provides ground for further research and outlines the path for the field to achieve a computational turn.
We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of difusion processes. Under suitable ergodicity assumptions on the underlying difusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense.