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We present a review of the history and the present state of the fractal approach to the large-scale distribution of galaxies. Angular correlation function was used as a general instrument for the structure analysis. It was realized later that a normalization condition for the reduced correlation function estimator results in distorted values for both R_{hom} and fractal dimension D. Moreover, according to a theorem on projections of fractals, galaxy angular catalogues can not be used for detecting a structure with the fractal dimension D>2. For this 3-d maps are required, and indeed modern extensive redshift-based 3-d maps have revealed the ``hidden fractal dimension of about 2, and have confirmed superclustering at scales even up to 500 Mpc (e.g. the Sloan Great Wall). On scales, where the fractal analysis is possible in completely embedded spheres, a power--law density field has been found. The fractal dimension D =2.2 +- 0.2 was directly obtained from 3-d maps and R_{hom} has expanded from 10 Mpc to scales approaching 100 Mpc. In concordance with the 3-d map results, modern all sky galaxy counts in the interval 10^m - 15^m give a 0.44m-law which corresponds to D=2.2 within a radius of 100h^{-1}_{100} Mpc. We emphasize that the fractal mass--radius law of galaxy clustering has become a key phenomenon in observational cosmology.
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The evolution and spatial structure of displacement fronts in fractures with self-affine rough walls are studied by numerical simulations. The fractures are open and the two faces are identical but shifted along their mean plane, either parallel or perpendicular to the flow. An initially flat front advected by the flow is progressively distorted into a self-affine front with Hurst exponent equal to that of the fracture walls. The lower cutoff of the self-affine regime depends on the aperture and lateral shift, while the upper cutoff grows linearly with the width of the front.
Self similar 3D distributions of point-particles, with a given quasifractal dimension D, were generated on a Menger sponge model and then compared with textit{2dfGRS} and textit{Virgo project} data footnote{http://www.mso.anu.edu.au/2dFGRS/, http://www.mpa-garching.mpg.de/Virgo/}. Using the principle of local knowledge, it is argued that in a finite volume of space only the two-point minus estimator is acceptable in the correlation analysis of self similar spatial distributions. In this sense, we have simplified the Pietronero-Labini correlative analysis by defining a K-minus estimator, which when applied to 2dfGRS data revealed the quasifractal dimension $Dapprox 2$ as expected. In our approach the K-minus estimator is used only locally. Dimensions between D = 1 and D = 1.7, as suggested by the standard $xi (r)$ analysis, were found to be fallacy of the method. In order to visualize spatial quasifractal objects, we created a small software program called textit{RoPo} (Rotate Points). This program illustrates and manifests local correlative analysis in which the visual inspection emerged as a first step and a key part of the method. Finally, we discuss importance and perspective of the visual inspection on available real and simulated distributions. It is also argued that results of contemporary cosmological simulations do not faithfully represent real data, as they show a formation of ever increasing collapsars. We consent that 2dfGRS data are reminiscent of some kind of underlying turbulence like effects in action.
We develop a field-theoretic description of large-scale structure formation by taking the non-relativistic limit of a canonically transformed, real scalar field which is minimally coupled to scalar gravitational perturbations in longitudinal gauge. We integrate out the gravitational constraint fields and arrive at a non-local action which is only specified in terms of the dynamical degrees of freedom. In order to make this framework closer to the classical particle description, we construct the corresponding 2PI effective action truncated at two loop order for a non-squeezed state without field expectation values. We contrast the dynamical description of the coincident time phase-space density to the standard Vlasov description of cold dark matter particles and identify momentum and time scales at which linear perturbation theory will deviate from the standard evolution.
The field of connectomics faces unprecedented big data challenges. To reconstruct neuronal connectivity, automated pixel-level segmentation is required for petabytes of streaming electron microscopy data. Existing algorithms provide relatively good accuracy but are unacceptably slow, and would require years to extract connectivity graphs from even a single cubic millimeter of neural tissue. Here we present a viable real-time solution, a multi-pass pipeline optimized for shared-memory multicore systems, capable of processing data at near the terabyte-per-hour pace of multi-beam electron microscopes. The pipeline makes an initial fast-pass over the data, and then makes a second slow-pass to iteratively correct errors in the output of the fast-pass. We demonstrate the accuracy of a sparse slow-pass reconstruction algorithm and suggest new methods for detecting morphological errors. Our fast-pass approach provided many algorithmic challenges, including the design and implementation of novel shallow convolutional neural nets and the parallelization of watershed and object-merging techniques. We use it to reconstruct, from image stack to skeletons, the full dataset of Kasthuri et al. (463 GB capturing 120,000 cubic microns) in a matter of hours on a single multicore machine rather than the weeks it has taken in the past on much larger distributed systems.
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Marie-Noelle Celerier (Laboratoire Univers et Theories
, LUTH
, n Observatoire de Paris-Meudon
2005
The structure of the large scale distribution of the galaxies have been widely studied since the publication of the first catalogs. Since large redshift samples are available, their analyses seem to show fractal correlations up to the observational limits. The value of the fractal dimension(s) calculated by different authors have become the object of a large debate, as have been the value of the expected transition from fractality to a possible large scale homogeneity. Moreover, some authors have proposed that different scaling regimes might be discerned at different lenght scales. To go further on into this issue, we have applied the correlation integral method to the wider sample currently available. We therefore obtain a fractal dimension of the galaxy distribution which seems to vary by steps whose width might be related to the organization hierarchy observed for the galaxies. This result could explain some of the previous results obtained by other authors from the analyses of less complete catalogs and maybe reconcile their apparent discrepancy. However, the method applied here needs to be further checked, since it produces odd fluctuations at each transition scale, which need to be thoroughly explained.
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