42,033 research outputs found
Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of convolutional neural networks.
Inspired by recent interest in geometric deep learning, which aims to
generalize convolutional neural networks to manifold and graph-structured
domains, we define a geometric scattering transform on manifolds. Similar to
the Euclidean scattering transform, the geometric scattering transform is based
on a cascade of wavelet filters and pointwise nonlinearities. It is invariant
to local isometries and stable to certain types of diffeomorphisms. Empirical
results demonstrate its utility on several geometric learning tasks. Our
results generalize the deformation stability and local translation invariance
of Euclidean scattering, and demonstrate the importance of linking the used
filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer
comment
Shape in an Atom of Space: Exploring quantum geometry phenomenology
A phenomenology for the deep spatial geometry of loop quantum gravity is
introduced. In the context of a simple model, an atom of space, it is shown how
purely combinatorial structures can affect observations. The angle operator is
used to develop a model of angular corrections to local, continuum flat-space
3-geometries. The physical effects involve neither breaking of local Lorentz
invariance nor Planck scale suppression, but rather reply on only the
combinatorics of SU(2) recoupling. Bhabha scattering is discussed as an example
of how the effects might be observationally accessible.Comment: 14 pages, 7 figures; v2 references adde
Calculation of quasi-degenerate energy levels of two-electron ions
Accurate QED calculations of the interelectron interaction corrections for
the , two-electron configurations for ions
with nuclear charge numbers are performed within the line
profile approach. Total energies of these configurations are evaluated.
Employing the fully relativistic treatment based on the {--} coupling
scheme these energy levels become quasi-degenerate in the region . To
treat such states within the framework of QED we utilize the line profile
approach. The calculations are performed within the Coulomb gauge.Comment: 22 pages, 11 figure
Spectra of graph neighborhoods and scattering
Let be a family of '-thin' Riemannian
manifolds modeled on a finite metric graph , for example, the
-neighborhood of an embedding of in some Euclidean space with
straight edges. We study the asymptotic behavior of the spectrum of the
Laplace-Beltrami operator on as , for various
boundary conditions. We obtain complete asymptotic expansions for the th
eigenvalue and the eigenfunctions, uniformly for , in
terms of scattering data on a non-compact limit space. We then use this to
determine the quantum graph which is to be regarded as the limit object, in a
spectral sense, of the family .
Our method is a direct construction of approximate eigenfunctions from the
scattering and graph data, and use of a priori estimates to show that all
eigenfunctions are obtained in this way.Comment: 37 pages, 3 figures, added references, added comment at end of
Section 1.2, changed comment after Theorem 30; in v4: made appendix into a
separate paper (arXiv:0711.2869), added reference, minor correction
Mathematics at the eve of a historic transition in biology
A century ago physicists and mathematicians worked in tandem and established
quantum mechanism. Indeed, algebras, partial differential equations, group
theory, and functional analysis underpin the foundation of quantum mechanism.
Currently, biology is undergoing a historic transition from qualitative,
phenomenological and descriptive to quantitative, analytical and predictive.
Mathematics, again, becomes a driving force behind this new transition in
biology.Comment: 5 pages, 2 figure
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