10,728 research outputs found
What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics
The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions
Metrics for generalized persistence modules
We consider the question of defining interleaving metrics on generalized
persistence modules over arbitrary preordered sets. Our constructions are
functorial, which implies a form of stability for these metrics. We describe a
large class of examples, inverse-image persistence modules, which occur
whenever a topological space is mapped to a metric space. Several standard
theories of persistence and their stability can be described in this framework.
This includes the classical case of sublevelset persistent homology. We
introduce a distinction between `soft' and `hard' stability theorems. While our
treatment is direct and elementary, the approach can be explained abstractly in
terms of monoidal functors.Comment: Final version; no changes from previous version. Published online Oct
2014 in Foundations of Computational Mathematics. Print version to appea
A Homology Theory for Etale Groupoids
Etale groupoids arise naturally as models for leaf spaces of foliations, for
orbifolds, and for orbit spaces of discrete group actions. In this paper we
introduce a sheaf homology theory for etale groupoids. We prove its invariance
under Morita equivalence, as well as Verdier duality between Haefliger
cohomology and this homology. We also discuss the relation to the cyclic and
Hochschild homologies of Connes' convolution algebra of the groupoid, and
derive some spectral sequences which serve as a tool for the computation of
these homologies.Comment: 34 page
Resolving Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations
Human brain anatomy and function display a combination of modular and
hierarchical organization, suggesting the importance of both cohesive
structures and variable resolutions in the facilitation of healthy cognitive
processes. However, tools to simultaneously probe these features of brain
architecture require further development. We propose and apply a set of methods
to extract cohesive structures in network representations of brain connectivity
using multi-resolution techniques. We employ a combination of soft
thresholding, windowed thresholding, and resolution in community detection,
that enable us to identify and isolate structures associated with different
weights. One such mesoscale structure is bipartivity, which quantifies the
extent to which the brain is divided into two partitions with high connectivity
between partitions and low connectivity within partitions. A second,
complementary mesoscale structure is modularity, which quantifies the extent to
which the brain is divided into multiple communities with strong connectivity
within each community and weak connectivity between communities. Our methods
lead to multi-resolution curves of these network diagnostics over a range of
spatial, geometric, and structural scales. For statistical comparison, we
contrast our results with those obtained for several benchmark null models. Our
work demonstrates that multi-resolution diagnostic curves capture complex
organizational profiles in weighted graphs. We apply these methods to the
identification of resolution-specific characteristics of healthy weighted graph
architecture and altered connectivity profiles in psychiatric disease.Comment: Comments welcom
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