2,834 research outputs found
Self-shrinkers with a rotational symmetry
In this paper we present a new family of non-compact properly embedded,
self-shrinking, asymptotically conical, positive mean curvature ends
that are hypersurfaces of revolution with
circular boundaries. These hypersurface families interpolate between the plane
and half-cylinder in , and any rotationally symmetric
self-shrinking non-compact end belongs to our family. The proofs involve the
global analysis of a cubic-derivative quasi-linear ODE. We also prove the
following classification result: a given complete, embedded, self-shrinking
hypersurface of revolution is either a hyperplane ,
the round cylinder of radius , the
round sphere of radius , or is diffeomorphic to an (i.e. a "doughnut" as in [Ang], which when is a torus). In
particular for self-shrinkers there is no direct analogue of the Delaunay
unduloid family. The proof of the classification uses translation and rotation
of pieces, replacing the method of moving planes in the absence of isometries.Comment: Trans. Amer. Math. Soc. (2011), to appear; 23 pages, 1 figur
Mean curvature self-shrinkers of high genus: Non-compact examples
We give the first rigorous construction of complete, embedded self-shrinking
hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The
surfaces exist for any sufficiently large prescribed genus , and are
non-compact with one end. Each has symmetries and comes from
desingularizing the intersection of the plane and sphere through a great
circle, a configuration with very high symmetry. Each is at infinity asymptotic
to the cone in over a -periodic graph on an equator
of the unit sphere , with the shape of a
periodically "wobbling sheet". This is a dramatic instability phenomenon, with
changes of asymptotics that break much more symmetry than seen in minimal
surface constructions. The core of the proof is a detailed understanding of the
linearized problem in a setting with severely unbounded geometry, leading to
special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the
gradient terms. This involves identifying new, adequate weighted H\"older
spaces of asymptotically conical functions in which the operators invert, via a
Liouville-type result with precise asymptotics.Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew.
Mat
Non-principal ultrafilters, program extraction and higher order reverse mathematics
We investigate the strength of the existence of a non-principal ultrafilter
over fragments of higher order arithmetic.
Let U be the statement that a non-principal ultrafilter exists and let
ACA_0^{\omega} be the higher order extension of ACA_0. We show that
ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that
ACA_0^{\omega}+\U is conservative over PA.
Moreover, we provide a program extraction method and show that from a proof
of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U
a realizing term in G\"odel's system T can be extracted. This means that one
can extract a term t, such that A(f,t(f))
On-site domestic wastewater renovation utilizing a partially saturated recirculating sand filter with lawn irrigation of effluent
A prototype on-site domestic wastewater renovation system was installed in 1988 and monitored during the period of February 1988 to November 1988. The prototype system used a partially saturated recirculating sand filter for renovation and a lawn irrigation sprinkler for distribution of renovated wastewater. The prototype was evaluated on the basis of its ability to remove nitrogen, COD, and coliform bacteria. A lawn plot receiving renovated wastewater, a potable-water irrigated plot and a dry control plot were sampled to evaluate the potential environmental impact of applying renovated wastewater to a lawn.
Water quality samples were collected weekly from various points within the prototype system. Runoff samples were collected from the lawn plots whenever natural precipitation events produced runoff. All water samples were analyzed for the concentration of nitrogen, COD, and coliform bacteria. Seasonal grass samples from the plots were checked for coliform bacterial content in an attempt to assess the potential for human exposure to pathogens by contact with and/or ingestion of lawn grass.
The investigation showed that the prototype had removal efficiencies as high as 82.5% for nitrogen, 96.3% for COD, and log reductions of 2.8 and 2.4 for fecal and total coliforms, respectively. Statistical comparisons of bacterial exposure, runoff water quality and grass yield among each of the plots showed few significant adverse differences when comparing the plot receiving renovated wastewater to the irrigated and dry control plots
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