The persistent cosmic web and its filamentary structure II: Illustrations

The recently introduced discrete persistent structure extractor (DisPerSE, Soubie 2010, paper I) is implemented on realistic 3D cosmological simulations and observed redshift catalogues (SDSS); it is found that DisPerSE traces equally well the observed filaments, walls, and voids in both cases. In either setting, filaments are shown to connect onto halos, outskirt walls, which circumvent voids. Indeed this algorithm operates directly on the particles without assuming anything about the distribution, and yields a natural (topologically motivated) self-consistent criterion for selecting the significance level of the identified structures. It is shown that this extraction is possible even for very sparsely sampled point processes, as a function of the persistence ratio. Hence astrophysicists should be in a position to trace and measure precisely the filaments, walls and voids from such samples and assess the confidence of the post-processed sets as a function of this threshold, which can be expressed relative to the expected amplitude of shot noise. In a cosmic framework, this criterion is comparable to friend of friend for the identifications of peaks, while it also identifies the connected filaments and walls, and quantitatively recovers the full set of topological invariants (Betti numbers) {\sl directly from the particles} as a function of the persistence threshold. This criterion is found to be sufficient even if one particle out of two is noise, when the persistence ratio is set to 3-sigma or more. The algorithm is also implemented on the SDSS catalogue and used to locat interesting configurations of the filamentary structure. In this context we carried the identification of an ``optically faint'' cluster at the intersection of filaments through the recent observation of its X-ray counterpart by SUZAKU. The corresponding filament catalogue will be made available online.Comment: A higher resolution version is available at http://www.iap.fr/users/sousbie together with complementary material (movie and data). Submitted to MNRA

Archives of Data Science, Series A. Vol. 1,1: Special Issue: Selected Papers of the 3rd German-Polish Symposium on Data Analysis and Applications

The first volume of Archives of Data Science, Series A is a special issue of a selection of contributions which have been originally presented at the {\em 3rd Bilateral German-Polish Symposium on Data Analysis and Its Applications} (GPSDAA 2013). All selected papers fit into the emerging field of data science consisting of the mathematical sciences (computer science, mathematics, operations research, and statistics) and an application domain (e.g. marketing, biology, economics, engineering)

Autoparallelity of Quantum Statistical Manifolds in The Light of Quantum Estimation Theory

In this paper we study the autoparallelity w.r.t. the e-connection for an information-geometric structure called the SLD structure, which consists of a Riemannian metric and mutually dual e- and m-connections, induced on the manifold of strictly positive density operators. Unlike the classical information geometry, the e-connection has non-vanishing torsion, which brings various mathematical difficulties. The notion of e-autoparallel submanifolds is regarded as a quantum version of exponential families in classical statistics, which is known to be characterized as statistical models having efficient estimators (unbiased estimators uniformly achieving the equality in the Cramer-Rao inequality). As quantum extensions of this classical result, we present two different forms of estimation-theoretical characterizations of the e-autoparallel submanifolds. We also give several results on the e-autoparallelity, some of which are valid for the autoparallelity w.r.t. an affine connection in a more general geometrical situation

Macro-permeability distribution and anisotropy in a 3D fissured and fractured clay rock: ‘Excavation Damaged Zone’ around a cylindrical drift in Callovo-Oxfordian Argilite (Bure)

The Underground Research Laboratory at Bure (CMHM), operated by ANDRA, the French National Radioactive Waste Management Agency, was developed for studying the disposal of radioactive waste in a deep clayey geologic repository. It comprises a network of underground galleries in a 130 m thick layer of Callovo Oxfordian clay rock (depths 400–600 m). This work focuses on hydraulic homogenization (permeability upscaling) of the Excavation Damaged Zone (EDZ) around a cylindrical drift, taking into account: (1) the permeability of the intact porous rock matrix; (2) the geometric structure of micro-fissures and small fractures synthesized as a statistical set of planar discs; (3) the curved shapes of large ‘chevron’ fractures induced by excavation (periodically distributed). The method used for hydraulic homogenization (upscaling) of the 3D porous and fractured rock is based on a ‘frozen gradient’ superposition of individual fluxes pertaining to each fracture/matrix block, or ‘unit block’. Each unit block comprises a prismatic block of permeable matrix (intact rock) obeying Darcy’s law, crossed by a single piece of planar fracture obeying either Darcy or Poiseuille law. Polygonal as well as disc shaped fractures are accommodated. The result of upscaling is a tensorial Darcy law, with macro-permeability Kij(x) distributed over a grid of upscaling sub-domains, or ‘voxels’. Alternatively, Kij(x) can be calculated point-wise using a moving window, e.g., for obtaining permeability profiles along ‘numerical’ boreholes. Because the permeable matrix is taken into account, the upscaling procedure can be implemented sequentially, as we do here: first, we embed the statistical fissures in the matrix, and secondly, we embed the large curved chevron fractures. The results of hydraulic upscaling are expressed first in terms of ‘equivalent’ macro-permeability tensors, Kij(x,y,z) distributed around the drift. The statistically isotropic fissures are considered, first, without chevron fractures. There are 10,000 randomly isotropic fissures distributed over a 20 m stretch of drift. The resulting spatially distributed K ij tensor is nearly isotropic (as expected). At the scale of the whole EDZ, the global K FISSURES is roughly 5000 times larger than permeability matrix KM. The detailed distribution of the equivalent K FISSURES (x, y, z) defined on a grid of voxels is radially inhomogeneous, like the statistics of the disc fissures. In addition, a moving window procedure is used to compute detailed radial profiles of K FISSURES versus distance (r) to drift wall, and the results compare favorably with in situ permeability profiles (numerical vs. experimental boreholes at Bure’s GMR drift). Finally, including the large curved chevron fractures in addition to the random fissures, the resulting K ij (x, y, z) appears strongly anisotropic locally. Its principal directions are spatially variable, and they tend to be aligned with the tangent planes of the chevron fracture surfaces. The global equivalent Kij of the whole EDZ is also obtained: it is only weakly anisotropic, much less so than the local Kij’s. However, because of the radially divergent structure of the ‘chevrons’ (although not quite cylindrical in geometry), it is recognized that the global Kij due to chevrons lacks physical meaning as a tensor. Considering only the magnitude, it is found that the permeability due to ‘chevrons’ (K CHEVRONS ) is about 4 orders of magnitude larger than that due to statistical fissures (K FISSURES ), assuming a hydraulic aperture a CHEVRON = 100 microns. By a simple argument, K CHEVRONS would be only one order of magnitude larger than K FISSURES with the choice a CHEVRON = 10 microns instead of 100 microns. This significant sensitivity is due to several factors: the large extent of chevron fractures, the assumption of constant hydraulic aperture, and the cubic law behavior based on the assumption of Poiseuille flow. The equivalent macro-permeabilities obtained in this work can be used for large scale flow modeling using any simulation code that accommodates Darcy’s law with a full, spatially variable permeability tensor Kij(x)

Information Geometry of Reversible Markov Chains

We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels

Using Persistent Homology for Topological Analysis of Protein Interaction Network of Candida Antarctica Lipase B Molecular Dynamic Simulation Model

In this work, we aim to examine the activity of one of the most efficient and commonly used lipases, Candida Antarctica Lipase B (CalB), from the perspective of multiple computational techniques. To this end, we first conduct a series of Molecular Dynam- ics Simulations on CalB in different conditions to analyze the conformational changes of the protein and probe its unusual high-temperature activity. Next, we build the protein interaction network of amino acids for CalB to study pairwise interactions between amino acids (nodes) and probe the protein in terms of statistical features of links’ distribution. Finally, we employ an algebraic topology-based method to study the protein interaction network from a broader perspective. The ”Persistent Homol- ogy (PH) method” is then presented as a way to exceed pairwise interactions and examine protein networks in terms of patterns of interaction between the nodes. Per- sistent Homology studies the evolution of the protein interaction network’s topologi- cal features (homology groups) in different states. Employing topological analysis, we compare the active form of CalB at high temperatures to its inactive states to account for possible topological contributions to the protein functionality. By discovering a prominent 1-dimensional hole in the active form of the protein, we highlight the role of higher-order interaction patterns in the network. Moreover, using the evolution of topological features, we study topological changes in protein networks and show the decline in the total number of 1-dimensional features as the protein loses activity and compactness over time. Accordingly, we propose that the protein’s general conforma- tional changes and three-dimensional structure are not the only facets contributing to its active state. Instead, we suggest examining the topology of the protein inter- action network, referred to as different dimensional holes of the networks, as a higher dimensional analysis should be used to account for protein functionality. Hence, in this work, we desire to present that one needs to consider topological features acting as patterns of interaction between the components to study, examine or predict the folding of polypeptide chains into active structures