30,360 research outputs found
Dilations for Systems of Imprimitivity acting on Banach Spaces
Motivated by a general dilation theory for operator-valued measures, framings
and bounded linear maps on operator algebras, we consider the dilation theory
of the above objects with special structures. We show that every
operator-valued system of imprimitivity has a dilation to a probability
spectral system of imprimitivity acting on a Banach space. This completely
generalizes a well-kown result which states that every frame representation of
a countable group on a Hilbert space is unitarily equivalent to a
subrepresentation of the left regular representation of the group. The dilated
space in general can not be taken as a Hilbert space. However, it can be taken
as a Hilbert space for positive operator valued systems of imprimitivity. We
also prove that isometric group representation induced framings on a Banach
space can be dilated to unconditional bases with the same structure for a
larger Banach space This extends several known results on the dilations of
frames induced by unitary group representations on Hilbert spaces.Comment: 21 page
Dilations of frames, operator valued measures and bounded linear maps
We will give an outline of the main results in our recent AMS Memoir, and
include some new results, exposition and open problems. In that memoir we
developed a general dilation theory for operator valued measures acting on
Banach spaces where operator-valued measures (or maps) are not necessarily
completely bounded. The main results state that any operator-valued measure,
not necessarily completely bounded, always has a dilation to a
projection-valued measure acting on a Banach space, and every bounded linear
map, again not necessarily completely bounded, on a Banach algebra has a
bounded homomorphism dilation acting on a Banach space. Here the dilation space
often needs to be a Banach space even if the underlying space is a Hilbert
space, and the projections are idempotents that are not necessarily
self-adjoint. These results lead to some new connections between frame theory
and operator algebras, and some of them can be considered as part of the
investigation about "noncommutative" frame theory.Comment: Contemporary Mathematics, 21 pages. arXiv admin note: substantial
text overlap with arXiv:1110.583
Clustered Graph Coloring and Layered Treewidth
A graph coloring has bounded clustering if each monochromatic component has
bounded size. This paper studies clustered coloring, where the number of colors
depends on an excluded complete bipartite subgraph. This is a much weaker
assumption than previous works, where typically the number of colors depends on
an excluded minor. This paper focuses on graph classes with bounded layered
treewidth, which include planar graphs, graphs of bounded Euler genus, graphs
embeddable on a fixed surface with a bounded number of crossings per edge,
amongst other examples. Our main theorem says that for fixed integers ,
every graph with layered treewidth at most and with no subgraph
is -colorable with bounded clustering. In the case, which
corresponds to graphs of bounded maximum degree, we obtain polynomial bounds on
the clustering. This greatly improves a corresponding result of Esperet and
Joret for graphs of bounded genus. The case implies that every graph with
a drawing on a fixed surface with a bounded number of crossings per edge is
5-colorable with bounded clustering. Our main theorem is also a critical
component in two companion papers that study clustered coloring of graphs with
no -subgraph and excluding a fixed minor, odd minor or topological
minor
Phaseless computational imaging with a radiating metasurface
Computational imaging modalities support a simplification of the active
architectures required in an imaging system and these approaches have been
validated across the electromagnetic spectrum. Recent implementations have
utilized pseudo-orthogonal radiation patterns to illuminate an object of
interest---notably, frequency-diverse metasurfaces have been exploited as fast
and low-cost alternative to conventional coherent imaging systems. However,
accurately measuring the complex-valued signals in the frequency domain can be
burdensome, particularly for sub-centimeter wavelengths. Here, computational
imaging is studied under the relaxed constraint of intensity-only measurements.
A novel 3D imaging system is conceived based on 'phaseless' and compressed
measurements, with benefits from recent advances in the field of phase
retrieval. In this paper, the methodology associated with this novel principle
is described, studied, and experimentally demonstrated in the microwave range.
A comparison of the estimated images from both complex valued and phaseless
measurements are presented, verifying the fidelity of phaseless computational
imaging.Comment: 18 pages, 18 figures, articl
The Mason Test: A Defense Against Sybil Attacks in Wireless Networks Without Trusted Authorities
Wireless networks are vulnerable to Sybil attacks, in which a malicious node
poses as many identities in order to gain disproportionate influence. Many
defenses based on spatial variability of wireless channels exist, but depend
either on detailed, multi-tap channel estimation - something not exposed on
commodity 802.11 devices - or valid RSSI observations from multiple trusted
sources, e.g., corporate access points - something not directly available in ad
hoc and delay-tolerant networks with potentially malicious neighbors. We extend
these techniques to be practical for wireless ad hoc networks of commodity
802.11 devices. Specifically, we propose two efficient methods for separating
the valid RSSI observations of behaving nodes from those falsified by malicious
participants. Further, we note that prior signalprint methods are easily
defeated by mobile attackers and develop an appropriate challenge-response
defense. Finally, we present the Mason test, the first implementation of these
techniques for ad hoc and delay-tolerant networks of commodity 802.11 devices.
We illustrate its performance in several real-world scenarios
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