Coarse Geometry of Topological Groups

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

Field-free topological behavior in the magnetic domain wall of ferrimagnetic GdFeCo

Exploring and controlling topological textures such as merons and skyrmions has attracted enormous interests from the perspective of fundamental research and spintronic applications. It has been predicted theoretically and proved experimentally that the lattice form of topological meron-skyrmion transformation can be realized with the requirement of external magnetic fields in chiral ferromagnets. However, such topological transition behavior has yet to be verified in other materials. Here, we report real-space observation of magnetic topology transformation between meron pairs and skyrmions in the localized domain wall of ferrimagnetic GdFeCo films without the need of magnetic fields. The topological transformation in the domain wall of ferrimagnet is introduced by temperature-induced spin reorientation transition (SRT) and the underlying mechanism is revealed by micromagnetic simulations. The convenient electric-controlling topology transformation and driving motion along the confined domain wall is further anticipated, which will enable advanced application in magnetic devices.

Electrochemical topological transformation of polysiloxanes

Coupling reactions between polymers are an important class of chemical modifications for changing, enhancing, and tuning the properties of polymeric materials. In particular, transformation of polymer topologies based on efficient, facile and less wasted coupling reactions remains a significant challenge. Here, we report coupling reactions based on electrochemical oxidation of 2,4,5-triphenylimidazole into a 2,4,5-triphenylimidazolyl radical and its spontaneous dimerization into hexaarylbiimidazole. Based on this chemistry, electrochemical topological transformation (ETT) and electrochemical chain extension have been realized with siloxane-based oligomers and polymers. Moreover, this approach enables one step ETT of star-shaped poly(dimethyl siloxane)s (PDMSs) into network PDMSs, running in an ionic liquid solvent and requiring no purification steps.

Aggregated Gaze Data Visualization Using Contiguous Irregular Cartograms

Gaze data visualization constitutes one of the most critical processes during eye-tracking analysis. Considering that modern devices are able to collect gaze data in extremely high frequencies, the visualization of the collected aggregated gaze data is quite challenging. In the present study, contiguous irregular cartograms are used as a method to visualize eye-tracking data captured by several observers during the observation of a visual stimulus. The followed approach utilizes a statistical grayscale heatmap as the main input and, hence, it is independent of the total number of the recorded raw gaze data. Indicative examples, based on different parameters/conditions and heatmap grid sizes, are provided in order to highlight their influence on the final image of the produced visualization. Moreover, two analysis metrics, referred to as center displacement (CD) and area change (AC), are proposed and implemented in order to quantify the geometric changes (in both position and area) that accompany the topological transformation of the initial heatmap grids, as well as to deliver specific guidelines for the execution of the used algorithm. The provided visualizations are generated using open-source software in a geographic information system.

A Novel Planar Differential Koch Fractal Eddy Current Probe with Parallel Wound Topological Structure

Flexible planar eddy current probes are widely used to detect conductive components with complex surface. In this study, topological transformation is applied to design a differential Koch coil exciting eddy current probe. Two kinds of Koch exciting coils, Koch A and Koch B , were obtained by topological transformation from a three-dimensional differential exciting eddy current probe. Finite element model simulation is conducted to visualize the differences of eddy current distributions induced by the probes and get the defect signal. A detailed comparison is made among the two kinds of Koch eddy current probes and a circular eddy current probe by experiments. The experiments demonstrate that the sensitivity of the Koch A eddy current probe is higher than that of another two probes for detecting the defect which is shorter than the size of the probe. This work provides a novel method for improving the performance of eddy current probes in the coil structure design.

Topological Analysis of the Brainstem of the Australian Lungfish Neoceratodus forsteri

This paper presents a survey of the cell masses in the brainstem of the Australian lungfish <i>Neoceratodus forsteri</i>, based on<i></i>transversely cut Bodian-stained serial sections, supplemented by immunohistochemical data from the recent literature. This study is intended to serve a double purpose. First it concludes and completes a series of publications on the structure of the brainstem in representative species of all groups of anamniote vertebrates. Within the framework of this comparative program the cell masses in the brainstem and their positional relations are analyzed in the light of the Herrick-Johnston concept, according to which the brainstem nuclei are arranged in four longitudinal, functional zones or columns, the boundaries of which are marked by ventricular sulci. The procedure employed in this analysis essentially involves two steps: first, the cell masses and large individual cells are projected upon the ventricular surface, and next, the ventricular surface is flattened out, that is, subjected to a one-to-one continuous topological transformation [J Comp Neurol. 1974;156:255–267]. The second purpose of the present paper is to complement our mapping of the longitudinal zonal arrangement of the cell masses in the brainstem of <i>Neoceratodus</i>with a subdivision in transversely oriented neural segments. Five longitudinal rhombencephalic sulci – the sulcus medianus inferior, the sulcus intermedius ventralis, the sulcus limitans, the sulcus intermedius dorsalis and the sulcus medianus superior – and four longitudinal mesencephalic sulci – the sulcus tegmentalis medialis, the sulcus tegmentalis lateralis, the sulcus subtectalis and the sulcus lateralis mesencephali – could be distinguished. Two obliquely oriented grooves, present in the isthmic region – the sulcus isthmi dorsalis and ventralis – deviate from the overall longitudinal pattern of the other sulci. Although in <i>Neoceratodus</i> most neuronal perikarya are situated within a diffuse periventricular gray, 45 cell masses could be delineated. Ten of these are primary efferent or motor nuclei, eight are primary afferent or sensory centers, six are considered to be components of the reticular formation and the remaining 21 may be interpreted as “relay” nuclei. The topological analysis showed that in most of the rhombencephalon the gray matter is arranged in four longitudinal zones or areas, termed area ventralis, area intermedioventralis, area intermediodorsalis and area dorsalis. The sulcus intermedius ventralis, the sulcus limitans, and the sulcus intermedius dorsalis mark the boundaries between these morphological entities. These longitudinal zones coincide largely, but not entirely, with the functional columns of Herrick and Johnston. The most obvious incongruity is that the area intermediodorsalis contains, in addition to the viscerosensory nucleus of the solitary tract, several general somatosensory and special somatosensory centers. The isthmus region does not exhibit a clear morphological zonal pattern. The mesencephalon is divisible into a ventral, primarily motor zone and a dorsal somatosensory zone. The boundary between these zones is marked by the sulcus tegmentalis lateralis, which may be considered as an isolated rostral extremity of the sulcus limitans. The results of this study are summarized in a “classical” topological map, as well as in a “modernized” version of this map, in which neuromere borders are indicated.