Homological algebra with locally compact abelian groups

In this article we study locally compact abelian (LCA) groups from the viewpoint of derived categories, using that their category is quasi-abelian in the sense of J.-P. Schneiders. We define a well-behaved derived Hom-complex with values in the derived category of Hausdorff topological abelian groups. Furthermore we introduce a smallness condition for LCA groups and show that such groups have a natural tensor product and internal Hom which both admit derived versions.Comment: 18 pages, AMSLaTe

Nonlinear estimation for linear inverse problems with error in the operator

We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov classes.Comment: Published in at http://dx.doi.org/10.1214/009053607000000721 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Complexity of the Interlace Polynomial

We consider the two-variable interlace polynomial introduced by Arratia, Bollobas and Sorkin (2004). We develop graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except on one line, where it is trivially polynomial time computable, and four lines, where the complexity is still open. This solves a problem posed by Arratia, Bollobas and Sorkin (2004). In particular, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at any point except at 0.Comment: 18 pages, 1 figure; new graph transformation (adding cycles) solves some unknown points, error in the statement of the inapproximability result fixed; a previous version has appeared in the proceedings of STACS 200

Nonparametric estimation of scalar diffusions based on low frequency data

We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X_{n\Delta},n=0,1,...,N) in a suitable Sobolev norm, together with an estimation of its invariant density.Comment: Published at http://dx.doi.org/10.1214/009053604000000797 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Early stopping for statistical inverse problems via truncated SVD estimation

We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension $D$. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level $\widehat m\in\{1,\ldots,D\}$ only based on the knowledge of the first $\widehat m$ singular values and vectors. We analyse in detail whether sequential {\it early stopping} rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.Comment: slightly modified version. arXiv admin note: text overlap with arXiv:1606.0770

Jörg Dinkelaker / Matthias Herrle: Erziehungswissenschaftliche Videographie. Eine Einführung. Wiesbaden: VS Verlag für Sozialwissenschaften 2009 [Rezension]

Rezension von: Jörg Dinkelaker / Matthias Herrle: Erziehungswissenschaftliche Videographie. Eine Einführung. Wiesbaden: VS Verlag für Sozialwissenschaften 2009, 134 S

Young tectonic evolution of the Northern Alpine Foreland Basin, southern Germany, based on linking geomorphology and structural geology

The Northern Alpine Foreland Basin (NAFB) formed as a result of alpine continental collision in Oligocene time. Tectonic shortening has been slow since the Miocene, implying that the NAFB subsidence has come to rest. The scope of this thesis is to find evidence that yield information regarding the recent vertical tectonic activity and a potentially ongoing basin inversion. The identification of such an active inversion can be accomplished by studying the landscape evolution. This is possible, because the earth’s surface contains information about tectonic and erosional processes. In order to understand and quantify the underlying mechanisms forming the landscape, both surface and subsurface information can be linked. Especially when geophysical or similar data are insufficient, geomorphological analyses provide new insights. The NAFB is a low-strain sedimentary basin, where historic subsurface data confirm subsidence, while the present-day erosional relief indicates uplift. The basin has a long prospection history and geometries of basement faults are well known, but their link to topography is ambiguous and the landscape is regarded as climatically controlled. In this context, dense population and energy exploration call for a fundamental understanding, if the basin is uplifting and whether or not faults could potentially be reactivated. This thesis aims at characterizing geomorphotectonic indicators of uplift, potentially indicating NAFB inversion. Related to this process, vertical motion of a few hundred meters to a few kilometers is expected to have been taking place during the Plio- Pleistocene. My study addresses the Quaternary geomorphology and distribution of sediments across the NAFB. To assess a potential link between asymmetric valleys and underlying basement faults in the NE portion of the basin (Tertiary Hills region), I analyzed high-resolution space-borne imagery and carried out geologic fieldwork. Results in distinct geomorphological response of the NAFB are (a) regional scale erosion of sediments, (b) fluvial incision, (c) reactivation of documented faults and (d) regional scale tilting of paleo-geodetic markers. 1In the NE portion of the NAFB, the geometry of a detected high-angle geologic subsurface contact associated with a surface escarpment, potentially originated from young surface faulting, while landscape morphology is intensely overprinted by climatic processes and anthropogenic land use. The investigation of mid-Miocene coastal features of the Swabian Alb cliff line addresses the landscape evolution along the marginal area of the NAFB. The particular questions here are, if coastal outcrops are part of the same coast, and which mechanisms have caused their different present-day elevations. I investigated marine features in a geologic field survey and compiled stratigraphic data to infer the regional response to uplift. The results show that two sites represent temporally and spatially different coastal sections, and both have been intensely modified by multiple erosional processes. The data can be explained by invoking a combination of surface erosion and possible lithospheric scale uplift. My study of well data and geomorphic markers, to infer NAFB Quaternary sediment budgets, yields a remarkable erosion of sediments along the central E-W basin axis and fluvial networks in disequilibrium. I examined erosion rates across different timescales, using sediment yields from NAFB catchments and basin sediment flux. On the Pleistocene timescale, geomorphic indicators of fluvial systems are investigated while for the Holocene, archeological data provide constraints on local erosion. The results of this study reveal a distinct oval-shaped erosion pattern across the NAFB, which can explain the modern erosional relief of the region, and further implies active inversion of the basin. Similar results are derived from drainage pattern and river steepness estimates. The results of this thesis imply that the Quaternary landscape in the NAFB has been significantly influenced by tectonic activity. Further, the timing for the NAFB transition from subsidence to uplift is likely to originate at the Pliocene – Quaternary boundary, and continues. Therefore, the possibility of future fault reactivation, controlled by lithospheric scale uplift, cannot be ruled out

Young tectonic evolution of the Northern Alpine Foreland Basin, southern Germany, based on linking geomorphology and structural geology

The Northern Alpine Foreland Basin (NAFB) formed as a result of alpine continental collision in Oligocene time. Tectonic shortening has been slow since the Miocene, implying that the NAFB subsidence has come to rest. The scope of this thesis is to find evidence that yield information regarding the recent vertical tectonic activity and a potentially ongoing basin inversion. The identification of such an active inversion can be accomplished by studying the landscape evolution. This is possible, because the earth’s surface contains information about tectonic and erosional processes. In order to understand and quantify the underlying mechanisms forming the landscape, both surface and subsurface information can be linked. Especially when geophysical or similar data are insufficient, geomorphological analyses provide new insights. The NAFB is a low-strain sedimentary basin, where historic subsurface data confirm subsidence, while the present-day erosional relief indicates uplift. The basin has a long prospection history and geometries of basement faults are well known, but their link to topography is ambiguous and the landscape is regarded as climatically controlled. In this context, dense population and energy exploration call for a fundamental understanding, if the basin is uplifting and whether or not faults could potentially be reactivated. This thesis aims at characterizing geomorphotectonic indicators of uplift, potentially indicating NAFB inversion. Related to this process, vertical motion of a few hundred meters to a few kilometers is expected to have been taking place during the Plio- Pleistocene. My study addresses the Quaternary geomorphology and distribution of sediments across the NAFB. To assess a potential link between asymmetric valleys and underlying basement faults in the NE portion of the basin (Tertiary Hills region), I analyzed high-resolution space-borne imagery and carried out geologic fieldwork. Results in distinct geomorphological response of the NAFB are (a) regional scale erosion of sediments, (b) fluvial incision, (c) reactivation of documented faults and (d) regional scale tilting of paleo-geodetic markers. 1In the NE portion of the NAFB, the geometry of a detected high-angle geologic subsurface contact associated with a surface escarpment, potentially originated from young surface faulting, while landscape morphology is intensely overprinted by climatic processes and anthropogenic land use. The investigation of mid-Miocene coastal features of the Swabian Alb cliff line addresses the landscape evolution along the marginal area of the NAFB. The particular questions here are, if coastal outcrops are part of the same coast, and which mechanisms have caused their different present-day elevations. I investigated marine features in a geologic field survey and compiled stratigraphic data to infer the regional response to uplift. The results show that two sites represent temporally and spatially different coastal sections, and both have been intensely modified by multiple erosional processes. The data can be explained by invoking a combination of surface erosion and possible lithospheric scale uplift. My study of well data and geomorphic markers, to infer NAFB Quaternary sediment budgets, yields a remarkable erosion of sediments along the central E-W basin axis and fluvial networks in disequilibrium. I examined erosion rates across different timescales, using sediment yields from NAFB catchments and basin sediment flux. On the Pleistocene timescale, geomorphic indicators of fluvial systems are investigated while for the Holocene, archeological data provide constraints on local erosion. The results of this study reveal a distinct oval-shaped erosion pattern across the NAFB, which can explain the modern erosional relief of the region, and further implies active inversion of the basin. Similar results are derived from drainage pattern and river steepness estimates. The results of this thesis imply that the Quaternary landscape in the NAFB has been significantly influenced by tectonic activity. Further, the timing for the NAFB transition from subsidence to uplift is likely to originate at the Pliocene – Quaternary boundary, and continues. Therefore, the possibility of future fault reactivation, controlled by lithospheric scale uplift, cannot be ruled out