TY - JOUR A1 - Eichmair, Michael A1 - Metzger, Jan T1 - Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions JF - Inventiones mathematicae Y1 - 2013 U6 - https://doi.org/10.1007/s00222-013-0452-5 SN - 0020-9910 SN - 1432-1297 VL - 194 IS - 3 SP - 591 EP - 630 PB - Springer CY - New York ER - TY - CHAP A1 - Hafer, Jörg A1 - Kiy, Alexander ED - Buchem, Ilona ED - Graham, Attwell ED - Tur, Gemma T1 - The university-wide introduction of an ePortfolio system as transdisciplinary task BT - Results of an implementation process and perspectives on an optimized process model T2 - Proceedings of the PLE Conference 2013: Learning and Diversity in the Cities of the Future Y1 - 2013 SP - 363 EP - 373 PB - Logos CY - Berlin ER - TY - JOUR A1 - Schmidt, Hans-Jürgen T1 - The tetralogy of Birkhoff theorems JF - General relativity and gravitation N2 - We classify the existent Birkhoff-type theorems into four classes: first, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum field equation with can be expressed by the Schwarzschild metric; for , it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces; including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of 2-D pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to 2-D. KW - Birkhoff theorem KW - Einstein space KW - Isometry group Y1 - 2013 U6 - https://doi.org/10.1007/s10714-012-1478-5 SN - 0001-7701 VL - 45 IS - 2 SP - 395 EP - 410 PB - Springer CY - New York ER - TY - JOUR A1 - Schmidt, Hans-Jürgen T1 - The tetralogy of Birkhoff theorems N2 - We classify the existent Birkhoff-type theorems into four classes: First, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum field equation with Lambda = 0 can be expressed by the Schwarzschild metric; for Lambda unequal 0, it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces; including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of two- dimensional pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to two dimensions. Y1 - 2013 UR - http://arXiv.org/abs/1208.5237 SN - 0001-7701 ER - TY - JOUR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - The stochastic chafee-infante equation JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics N2 - In this preparatory chapter, the tools of stochastic analysis needed for the investigation of the asymptotic behavior of the stochastic Chafee-Infante equation are provided. In the first place, this encompasses a recollection of basic facts about Lévy processes with values in Hilbert spaces. Playing the role of the additive noise processes perturbing the deterministic Chafee-Infante equation in the systems the stochastic dynamics of which will be our main interest, symmetric ?-stable Lévy processes are in the focus of our investigation (Sect. 3.1). Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_3 SN - 0075-8434 VL - 2085 SP - 45 EP - 68 PB - Springer CY - Berlin ER - TY - JOUR A1 - Debussche, Arnaud A1 - Hoegele, Michael A1 - Imkeller, Peter T1 - The source of stochastic models in conceptual climate dynamics JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8 SN - 0075-8434 VL - 2085 IS - 3 SP - 151 EP - 157 PB - Springer CY - Berlin ER - TY - JOUR A1 - Debussche, Arnaud A1 - Hoegele, Michael A1 - Imkeller, Peter T1 - The small deviation of the small noise solution JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_4 SN - 0075-8434 VL - 2085 SP - 69 EP - 85 PB - Springer CY - Berlin ER - TY - JOUR A1 - Zöller, Gert A1 - Holschneider, Matthias A1 - Hainzl, Sebastian T1 - The Maximum Earthquake Magnitude in a Time Horizon: Theory and Case Studies JF - Bulletin of the Seismological Society of America N2 - We show how the maximum magnitude within a predefined future time horizon may be estimated from an earthquake catalog within the context of Gutenberg-Richter statistics. The aim is to carry out a rigorous uncertainty assessment, and calculate precise confidence intervals based on an imposed level of confidence a. In detail, we present a model for the estimation of the maximum magnitude to occur in a time interval T-f in the future, given a complete earthquake catalog for a time period T in the past and, if available, paleoseismic events. For this goal, we solely assume that earthquakes follow a stationary Poisson process in time with unknown productivity Lambda and obey the Gutenberg-Richter law in magnitude domain with unknown b-value. The random variables. and b are estimated by means of Bayes theorem with noninformative prior distributions. Results based on synthetic catalogs and on retrospective calculations of historic catalogs from the highly active area of Japan and the low-seismicity, but high-risk region lower Rhine embayment (LRE) in Germany indicate that the estimated magnitudes are close to the true values. Finally, we discuss whether the techniques can be extended to meet the safety requirements for critical facilities such as nuclear power plants. For this aim, the maximum magnitude for all times has to be considered. In agreement with earlier work, we find that this parameter is not a useful quantity from the viewpoint of statistical inference. Y1 - 2013 U6 - https://doi.org/10.1785/0120120013 SN - 0037-1106 VL - 103 IS - 2A SP - 860 EP - 875 PB - Seismological Society of America CY - Albany ER - TY - JOUR A1 - Shojaei-Fard, Ali T1 - The global beta-functions from solutions of dyson-schwinger equations JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We apply the geometric interpretation of Dyson-Schwinger equations (DSEs) in terms of equi-singular flat connections to provide a process which relates beta-functions of a DSE under different regularization schemes. KW - Dyson-Schwinger equations KW - beta-functions KW - renormalization Hopf algebra KW - equi-singular connections Y1 - 2013 U6 - https://doi.org/10.1142/S0217732313501526 SN - 0217-7323 SN - 1793-6632 VL - 28 IS - 34 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Debussche, Arnaud A1 - Högele, Michael A1 - Imkeller, Peter T1 - The Fine Dynamics of the Chafee-Infante Equation JF - Lecture notes in mathematics : a collection of informal reports and seminars JF - Lecture Notes in Mathematics N2 - In this chapter, we introduce the deterministic Chafee-Infante equation. This equation has been the subject of intense research and is very well understood now. We recall some properties of its longtime dynamics and in particular the structure of its attractor. We then define reduced domains of attraction that will be fundamental in our study and give a result describing precisely the time that a solution starting form a reduced domain of attraction needs to reach a stable equilibrium. This result is then proved using the detailed knowledge of the attractor and classical tools such as the stable and unstable manifolds in a neighborhood of an equilibrium. Y1 - 2013 SN - 978-3-319-00828-8; 978-3-319-00827-1 U6 - https://doi.org/10.1007/978-3-319-00828-8_2 SN - 0075-8434 VL - 2085 SP - 11 EP - 43 PB - Springer CY - Berlin ER -