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Abstract Given a lattice Veech group in the mapping class group of a closed surface , this paper investigates the geometry of , the associated ‐extension group. We prove that is the fundamental group of a bundle with a singular Euclidean‐by‐hyperbolic geometry. Our main result is that collapsing “obvious” product regions of the universal cover produces an action of on a hyperbolic space, retaining most of the geometry of . This action is a key ingredient in the sequel where we show that is hierarchically hyperbolic and quasi‐isometrically rigid.
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Gabor frames from contact geometry in models of the primary visual cortex
We analyze the interplay between contact geometry and Gabor filters signalanalysis in geometric models of the primary visual cortex. We show inparticular that a specific framed lattice and an associated Gabor system isdetermined by the Legendrian circle bundle structure of the $$3$$-manifold ofcontact elements on a surface (which models the V1-cortex), together with thepresence of an almost-complex structure on the tangent bundle of the surface(which models the retinal surface). We identify a scaling of the lattice, alsodictated by the manifold geometry, that ensures the frame condition issatisfied. We then consider a $$5$$-dimensional model where receptor profilesalso involve a dependence on frequency and scale variables, in addition to thedependence on position and orientation. In this case we show that a proposedprofile window function does not give rise to frames (even in a distributionalsense), while a natural modification of the same generates Gabor frames withrespect to the appropriate lattice determined by the contact geometry.
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- Award ID(s):
- 2104330
- PAR ID:
- 10598859
- Publisher / Repository:
- EPI Sciences
- Date Published:
- Journal Name:
- Mathematical Neuroscience and Applications
- Volume:
- Volume 3
- ISSN:
- 2801-0159
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.46298/mna.9766
