6,978 research outputs found
The Vietoris-Rips complexes of a circle
Given a metric space X and a distance threshold r>0, the Vietoris-Rips
simplicial complex has as its simplices the finite subsets of X of diameter
less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian
manifold and r is sufficiently small, then the Vietoris-Rips complex is
homotopy equivalent to the original manifold. Little is known about the
behavior of Vietoris-Rips complexes for larger values of r, even though these
complexes arise naturally in applications using persistent homology. We show
that as r increases, the Vietoris-Rips complex of the circle obtains the
homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ...,
until finally it is contractible. As our main tool we introduce a directed
graph invariant, the winding fraction, which in some sense is dual to the
circular chromatic number. Using the winding fraction we classify the homotopy
types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset
of the circle, and we study the expected homotopy type of the Vietoris-Rips
complex of a uniformly random sample from the circle. Moreover, we show that as
the distance parameter increases, the ambient Cech complex of the circle also
obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the
7-sphere, ..., until finally it is contractible.Comment: Final versio
Evasion Paths in Mobile Sensor Networks
Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't
know its location but does know when it overlaps a nearby sensor. We say that
an evasion path exists in this sensor network if a moving intruder can avoid
detection. In "Coordinate-free coverage in sensor networks with controlled
boundaries via homology", Vin deSilva and Robert Ghrist give a necessary
condition, depending only on the time-varying connectivity data of the sensors,
for an evasion path to exist. Using zigzag persistent homology, we provide an
equivalent condition that moreover can be computed in a streaming fashion.
However, no method with time-varying connectivity data as input can give
necessary and sufficient conditions for the existence of an evasion path.
Indeed, we show that the existence of an evasion path depends not only on the
fibrewise homotopy type of the region covered by sensors but also on its
embedding in spacetime. For planar sensors that also measure weak rotation and
distance information, we provide necessary and sufficient conditions for the
existence of an evasion path
On Vietoris-Rips complexes of ellipses
For a metric space and a scale parameter, the Vietoris-Rips complex
(resp. ) has as its vertex set, and a finite
subset as a simplex whenever the diameter of is
less than (resp. at most ). Though Vietoris-Rips complexes have been
studied at small choices of scale by Hausmann and Latschev, they are not
well-understood at larger scale parameters. In this paper we investigate the
homotopy types of Vietoris-Rips complexes of ellipses of small eccentricity, meaning
. Indeed, we show there are constants such that for
all , we have and , though only one of the two-spheres in is
persistent. Furthermore, we show that for any scale parameter ,
there are arbitrarily dense subsets of the ellipse such that the Vietoris-Rips
complex of the subset is not homotopy equivalent to the Vietoris-Rips complex
of the entire ellipse. As our main tool we link these homotopy types to the
structure of infinite cyclic graphs
The Federalist Regulation of Privacy: The Happy Incidents of State Regulatory Activity and Costs of Preemptive Federal Action
The impact of society’s digital integration is difficult to articulate. It suffices to say much of our lives are now digitized, and digital technologies have yielded immeasurable benefits to the individual and society at large. Change heralds challenge, and the digital paradigm-shift has brought challenges of comparable numerosity and magnitude. Privacy is at the forefront of those challenges. In recent years, the digital industry has been subject to increased scrutiny over the rising number of privacy scandals and perceived market failures related to the collection and use of individuals’ personal information. New technologies, market developments, and increases in public attention have culminated in widespread perceptions of privacy threats and abuses. Governments around the globe are responding by revamping their regulation of privacy and the digital industry. In stark contrast, the United States federal government has maintained its rudimental self-regulatory approach. A handful of states, spearheaded by California’s enaction of the California Consumer Privacy Act of 2018 (“CCPA”), have moved to fill the gap left by federal inaction. The scope of the CCPA is unrivalled by any previous United States privacy regulation, and with its activation date quickly approaching, industry actors have focused their lobbying efforts in Washington D.C. to the increasing reception of federal legislators. Any congressional action could have major repercussions for state and federal regulators’ ability to police the collection and use of citizens’ personal information, and accordingly, such action may redefine privacy in the United States. The present scenario raises important questions about federalism and novel informational privacy regulations. Few commentators have addressed the issue directly, and no one has done so recently. What role should the federal government and states play in addressing the privacy concerns of Americans? Should the federal government preempt the CCPA and its progeny in favor of active federal regulation of the digital industry’s collection and use of personal information? What are the consequences of allowing the CCPA and similar state laws to regulate the control of their citizens’ personal information? This Comment will explore such questions
The Development and Utilization of Seapower by Great Britain
Admiral Colbert, faculty and students of the School of Naval Warfare, I am very happy to have the opportunity to speak to you this morning on the subject of The Development and Utilization of Seapower by Greal Britain. The problem is where to begin, how to say anything significant in the space of time allotted to me, and, if possible, stop
Nerve complexes of circular arcs
We show that the nerve complex of n arcs in the circle is homotopy equivalent
to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the
same even dimension. Moreover this homotopy type can be computed in time O(n
log n). For the particular case of the nerve complex of evenly-spaced arcs of
the same length, we determine the dihedral group action on homology, and we
relate the complex to a cyclic polytope with n vertices. We give three
applications of our knowledge of the homotopy types of nerve complexes of
circular arcs. First, we use the connection to cyclic polytopes to give a novel
topological proof of a known upper bound on the distance between successive
roots of a homogeneous trigonometric polynomial. Second, we show that the
Lovasz bound on the chromatic number of a circular complete graph is either
sharp or off by one. Third, we show that the Vietoris--Rips simplicial complex
of n points in the circle is homotopy equivalent to either a point, an
odd-dimensional sphere, or a wedge sum of spheres of the same even dimension,
and furthermore this homotopy type can be computed in time O(n log n)
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