The effective longitudinal dispersion constant, D(L)(eff), in cylindrical packed beds is larger than in the bulk due to the existence of radial inhomogeneities induced by the cylinder walls. For dense random packed beds, D(L)(eff) can be several times larger than the bulk value, even for arbitrarily large cylinder radius, R. The time-scale for attaining asymptotic dispersion rates in a cylindrical geometry is neither the convective nor the diffusive time-scale, but rather D(T)/R(2), where D(T) is the bulk transverse dispersion rate. Similar effects are predicted for packed beds confined in ducts of any cross-sectional geometry. The case of a rectangular duct, compared with an infinite slit, provides an intuitive model for the influence of walls in the limit as R goes to infinity.
The term brain activity refers to a wide range of mental faculties that can be assessed either on anatomical/functional micro-, meso-and macro-spatiotemporal scales of observation, or on intertwined levels with mutual interactions. Here we show, based on novel topological findings, how every brain activity encompassed in a given anatomical or functional level necessarily displays a counterpart in others. Different brain activities are able to scatter, collide and combine, merging, condensing and stitching together in a testable and quantifiable way. We point out how, despite their local cortical functional differences, all the mental processes, from perception to emotions, from cognition to mind wandering, may be reduced to a single, general brain activity that takes place in dimensions higher than the classical 3D plus time. In physics, the term duality refers to the case where two seemingly different systems turn out to be equivalent. Our framework permits a topological duality among different brain activities and neuro-techniques, because it holds for all the types of spatio-temporal nervous functions, independent of their cortical location, inter-and intra-level relationships, strength, magnitude and boundaries.
The nervous activity of the brain takes place in higher-dimensional functional spaces. Indeed, recent claims advocate that the brain might be equipped with a phase space displaying four spatial dimensions plus time, instead of the classical three plus time. This suggests the possibility to investigate global visualization methods for exploiting four-dimensional maps of real experimental data sets. Here we asked whether, starting from the conventional neuro-data available in three dimensions plus time, it is feasible to find an operational procedure to describe the corresponding four-dimensional trajectories. In particular, we used quaternion orthographic projections for the assessment of electroencephalographic traces (EEG) from scalp locations. This approach makes it possible to map three-dimensional EEG traces to the surface of a four-dimensional hypersphere, which has an important advantage, since quaternionic networks make it feasible to enlighten temporally far apart nervous trajectories equipped with the same features, such as the same frequency or amplitude of electric oscillations. This leads to an incisive operational assessment of symmetries, dualities and matching descriptions hidden in the very structure of complex neuro-data signals.
Spatio-temporal brain activities with variable delay detectable in resting-state functional magnetic resonance imaging (rs-fMRI) give rise to highly reproducible structures, termed cortical lag threads, that can propagate from one brain region to another. Using a computational topology of data approach, we found that Betti numbers that are cycle counts and the areas of vortex cycles covering brain activation regions in triangulated rs-fMRI video frames make it possible to track persistent, recurring blood oxygen level dependent (BOLD) signals. Our findings have been codified and visualized in what are known as persistent barcodes. Importantly, a topology of data offers a practical approach in coping with and sidestepping massive noise in neuro data, such as unwanted dark (low intensity) regions in the neighbourhood of non-zero BOLD signals. A natural outcome of a topology of data approach is the tracking of persistent, non-trivial BOLD signals that appear intermittently in a sequence of rs-fMRI video frames. The end result of this tracking of changing lag structures is a persistent barcode, which is a pictograph that offers a convenient visual means of exhibiting, comparing and classifying brain activation patterns.
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