Sequential order under CH

Revisiting and completing a work due to A. I. Ba\v{s}kirov, we construct compact sequential spaces of any sequential order up to and including $\omega_1$ as quotient spaces of $\beta\omega$ under CH

The efficient certification of knottedness and Thurston norm

We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable 3-manifold is equal to a given integer is in NP. As a corollary, the problem of determining the genus of a knot in the 3-sphere is in NP. We also show that the problem of determining whether a compact orientable 3-manifold has incompressible boundary is in NP.Comment: 101 pages, 24 figures; v2 contains some improvements suggested by the referee, which have strengthened the main theorem

Safe Routes to School Improves the Built Environment

This report focuses on case studies describing how ten states (California, District of Columbia, Georgia, Illinois, Kentucky, Louisiana, New York, Oklahoma, Texas and Virginia) are awarding their SRTS federal funds to support improved infrastructure such as sidewalks, bike lanes, pathways, improved intersections, traffic calming, and more. Safe Routes to School: Improves the Built Environment shares information on local level implementation challenges, best practices, and securing more improvements to the built environment in your community

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur

The number of Reidemeister Moves Needed for Unknotting

There is a positive constant $c_1$ such that for any diagram $D$ representing the unknot, there is a sequence of at most $2^{c_1 n}$ Reidemeister moves that will convert it to a trivial knot diagram, $n$ is the number of crossings in $D$. A similar result holds for elementary moves on a polygonal knot $K$ embedded in the 1-skeleton of the interior of a compact, orientable, triangulated $PL$ 3-manifold $M$. There is a positive constant $c_2$ such that for each $t \geq 1$, if $M$ consists of $t$ tetrahedra, and $K$ is unknotted, then there is a sequence of at most $2^{c_2 t}$ elementary moves in $M$ which transforms $K$ to a triangle contained inside one tetrahedron of $M$. We obtain explicit values for $c_1$ and $c_2$.Comment: 48 pages, 14 figure

Optics: general-purpose scintillator light response simulation code

We present the program optics that simulates the light response of an arbitrarily shaped scintillation particle detector. Predicted light responses of pure CsI polygonal detectors, plastic scintillator staves, cylindrical plastic target scintillators and a Plexiglas light-distribution plate are illustrated. We demonstrate how different bulk and surface optical properties of a scintillator lead to specific volume and temporal light collection probability distributions. High-statistics optics simulations are calibrated against the detector responses measured in a custom-made cosmic muon tomography apparatus. The presented code can also be used to track particles intersecting complex geometrical objects.Comment: RevTeX LaTeX, 37 pages in e-print format, 12 Postscript Figures and 1 Table, also available at http://pibeta.phys.virginia.edu/public_html/preprints/optics.p