Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

A g-factor metric for k-t-GRAPPA- and PEAK-GRAPPA-based parallel imaging.

PURPOSE The aim of this work is to derive a theoretical framework for quantitative noise and temporal fidelity analysis of time-resolved k-space-based parallel imaging methods. THEORY An analytical formalism of noise distribution is derived extending the existing g-factor formulation for nontime-resolved generalized autocalibrating partially parallel acquisition (GRAPPA) to time-resolved k-space-based methods. The noise analysis considers temporal noise correlations and is further accompanied by a temporal filtering analysis. METHODS All methods are derived and presented for k-t-GRAPPA and PEAK-GRAPPA. A sliding window reconstruction and nontime-resolved GRAPPA are taken as a reference. Statistical validation is based on series of pseudoreplica images. The analysis is demonstrated on a short-axis cardiac CINE dataset. RESULTS The superior signal-to-noise performance of time-resolved over nontime-resolved parallel imaging methods at the expense of temporal frequency filtering is analytically confirmed. Further, different temporal frequency filter characteristics of k-t-GRAPPA, PEAK-GRAPPA, and sliding window are revealed. CONCLUSION The proposed analysis of noise behavior and temporal fidelity establishes a theoretical basis for a quantitative evaluation of time-resolved reconstruction methods. Therefore, the presented theory allows for comparison between time-resolved parallel imaging methods and also nontime-resolved methods. Magn Reson Med, 2014. © 2014 Wiley Periodicals, Inc.

The hp-adaptive FEM based on continuous Sobolev embeddings: isotropic refinements

The aim of this paper is to present a new class of smoothness testing strategies in the context of hp-adaptive refinements based on continuous Sobolev embeddings. In addition to deriving a modified form of the 1d smoothness indicators introduced in [26], they will be extended and applied to a higher dimensional framework. A few numerical experiments in the context of the hp-adaptive FEM for a linear elliptic PDE will be performed.

VIRTOPSY--scientific documentation, reconstruction and animation in forensic: individual and real 3D data based geo-metric approach including optical body/object surface and radiological CT/MRI scanning.

Until today, most of the documentation of forensic relevant medical findings is limited to traditional 2D photography, 2D conventional radiographs, sketches and verbal description. There are still some limitations of the classic documentation in forensic science especially if a 3D documentation is necessary. The goal of this paper is to demonstrate new 3D real data based geo-metric technology approaches. This paper present approaches to a 3D geo-metric documentation of injuries on the body surface and internal injuries in the living and deceased cases. Using modern imaging methods such as photogrammetry, optical surface and radiological CT/MRI scanning in combination it could be demonstrated that a real, full 3D data based individual documentation of the body surface and internal structures is possible in a non-invasive and non-destructive manner. Using the data merging/fusing and animation possibilities, it is possible to answer reconstructive questions of the dynamic development of patterned injuries (morphologic imprints) and to evaluate the possibility, that they are matchable or linkable to suspected injury-causing instruments. For the first time, to our knowledge, the method of optical and radiological 3D scanning was used to document the forensic relevant injuries of human body in combination with vehicle damages. By this complementary documentation approach, individual forensic real data based analysis and animation were possible linking body injuries to vehicle deformations or damages. These data allow conclusions to be drawn for automobile accident research, optimization of vehicle safety (pedestrian and passenger) and for further development of crash dummies. Real 3D data based documentation opens a new horizon for scientific reconstruction and animation by bringing added value and a real quality improvement in forensic science.

Explora: A Visualisation Tool for Metric Analysis of Software Corpora

When analysing software metrics, users find that visualisation tools lack support for (1) the detection of patterns within metrics; and (2) enabling analysis of software corpora. In this paper we present Explora, a visualisation tool designed for the simultaneous analysis of multiple metrics of systems in software corpora. Explora incorporates a novel lightweight visualisation technique called PolyGrid that promotes the detection of graphical patterns. We present an example where we analyse the relation of subtype polymorphism with inheritance and invocation in corpora of Smalltalk and Java systems and find that (1) subtype polymorphism is more likely to be found in large hierarchies; (2) as class hierarchies grow horizontally, they also do so vertically; and (3) in polymorphic hierarchies the length of the name of the classes is orthogonal to the cardinality of the call sites.