1,692 research outputs found
Finite Voronoi decompositions of infinite vertex transitive graphs
In this paper, we consider the Voronoi decompositions of an arbitrary
infinite vertex-transitive graph G. In particular, we are interested in the
following question: what is the largest number of Voronoi cells that must be
infinite, given sufficiently (but finitely) many Voronoi sites which are
sufficiently far from each other? We call this number the survival number s(G).
The survival number of a graph has an alternative characterization in terms
of covering, which we use to show that s(G) is always at least two. The
survival number is not a quasi-isometry invariant, but it remains open whether
finiteness of the s(G) is. We show that all vertex transitive graphs with
polynomial growth have a finite s(G); vertex transitive graphs with infinitely
many ends have an infinite s(G); the lamplighter graph LL(Z), which has
exponential growth, has a finite s(G); and the lamplighter graph LL(Z^2), which
is Liouville, has an infinite s(G)
Good covers are algorithmically unrecognizable
A good cover in R^d is a collection of open contractible sets in R^d such
that the intersection of any subcollection is either contractible or empty.
Motivated by an analogy with convex sets, intersection patterns of good covers
were studied intensively. Our main result is that intersection patterns of good
covers are algorithmically unrecognizable.
More precisely, the intersection pattern of a good cover can be stored in a
simplicial complex called nerve which records which subfamilies of the good
cover intersect. A simplicial complex is topologically d-representable if it is
isomorphic to the nerve of a good cover in R^d. We prove that it is
algorithmically undecidable whether a given simplicial complex is topologically
d-representable for any fixed d \geq 5. The result remains also valid if we
replace good covers with acyclic covers or with covers by open d-balls.
As an auxiliary result we prove that if a simplicial complex is PL embeddable
into R^d, then it is topologically d-representable. We also supply this result
with showing that if a "sufficiently fine" subdivision of a k-dimensional
complex is d-representable and k \leq (2d-3)/3, then the complex is PL
embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in
version
Cutting the same fraction of several measures
We study some measure partition problems: Cut the same positive fraction of
measures in with a hyperplane or find a convex subset of
on which given measures have the same prescribed value. For
both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure
Robotic excision of a difficult retrorectal cyst â a video vignette
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/153653/1/codi14862_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/153653/2/codi14862.pd
Evidence of adaptation, niche separation and microevolution within the genus Polaromonas on Arctic and Antarctic glacial surfaces
Polaromonas is one of the most abundant genera found on glacier surfaces, yet it’s ecology remains poorly described. Investigations made to date point towards a uniform distribution of Polaromonas phylotypes across the globe. We compared 43 Polaromonas isolates obtained from surfaces of Arctic and Antarctic glaciers to address this issue. 16S rRNA gene sequences, intergenic transcribed spacers (ITS) and metabolic fingerprinting showed great differences between hemispheres but also between neighboring glaciers. Phylogenetic distance between Arctic and Antarctic isolates indicated separate species. The Arctic group clustered similarly, when constructing dendrograms based on 16S rRNA gene and ITS sequences, as well as metabolic traits. The Antarctic strains, although almost identical considering 16S rRNA genes, diverged into 2 groups based on the ITS sequences and metabolic traits, suggesting recent niche separation. Certain phenotypic traits pointed towardscell adaptation to specific conditions on a particular glacier, like varying pH levels. Collected data suggest, that seeding of glacial surfaces with Polaromonas cells transported by various means, is of greater efficiency on local than global scales. Selection mechanisms present of glacial surfaces reduce the deposited Polaromonas diversity, causing subsequent adaptation to prevailing environmental conditions. Furthermore, interactions with other supraglacial microbiota, like algae cells may drive postselectional niche separation and microevolution within the Polaromonas genus
How large are the level sets of the Takagi function?
Let T be Takagi's continuous but nowhere-differentiable function. This paper
considers the size of the level sets of T both from a probabilistic point of
view and from the perspective of Baire category. We first give more elementary
proofs of three recently published results. The first, due to Z. Buczolich,
states that almost all level sets (with respect to Lebesgue measure on the
range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states
that the average number of points in a level set is infinite. The third result,
also due to Lagarias and Maddock, states that the average number of local level
sets contained in a level set is 3/2. In the second part of the paper it is
shown that, in contrast to the above results, the set of ordinates y with
uncountably infinite level sets is residual, and a fairly explicit description
of this set is given. The paper also gives a negative answer to a question of
Lagarias and Maddock by showing that most level sets (in the sense of Baire
category) contain infinitely many local level sets, and that a continuum of
level sets even contain uncountably many local level sets. Finally, several of
the main results are extended to a version of T with arbitrary signs in the
summands.Comment: Added a new Section 5 with generalization of the main results; some
new and corrected proofs of the old material; 29 pages, 3 figure
Singular solutions of fully nonlinear elliptic equations and applications
We study the properties of solutions of fully nonlinear, positively
homogeneous elliptic equations near boundary points of Lipschitz domains at
which the solution may be singular. We show that these equations have two
positive solutions in each cone of , and the solutions are unique
in an appropriate sense. We introduce a new method for analyzing the behavior
of solutions near certain Lipschitz boundary points, which permits us to
classify isolated boundary singularities of solutions which are bounded from
either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as
well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure
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