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

Ionic and electronic properties of the topological insulator Bi2_2Te2_2Se investigated using β\beta-detected nuclear magnetic relaxation and resonance of 8^8Li

We report measurements on the high temperature ionic and low temperature electronic properties of the 3D topological insulator Bi$_2$Te$_2$Se using ion-implanted $^8$Li $\beta$-detected nuclear magnetic relaxation and resonance. With implantation energies in the range 5-28 keV, the probes penetrate beyond the expected range of the topological surface state, but are still within 250 nm of the surface. At temperatures above ~150 K, spin-lattice relaxation measurements reveal isolated $^8$Li$^{+}$ diffusion with an activation energy $E_{A} = 0.185(8)$ eV and attempt frequency $\tau_{0}^{-1} = 8(3) \times 10^{11}$ s$^{-1}$ for atomic site-to-site hopping. At lower temperature, we find a linear Korringa-like relaxation mechanism with a field dependent slope and intercept, which is accompanied by an anomalous field dependence to the resonance shift. We suggest that these may be related to a strong contribution from orbital currents or the magnetic freezeout of charge carriers in this heavily compensated semiconductor, but that conventional theories are unable to account for the extent of the field dependence. Conventional NMR of the stable host nuclei may help elucidate their origin.Comment: 17 pages, 12 figures, submitted to Phys. Rev.

Zum Stufenaufbau des Parallelenaxioms

Euclid 's parallel postulate is shown to be equivalent to the conjunction of the following two weaker postulates: “Any perpendicular to one side of a right angle intersects any perpendicular to the other side” and “For any acute angle Oxy, the segment PQ — where P is a point on O x , Q a point on O y and PQ ⊥ Oy — grows indefinitely, i. e. can be made longer than any given segment”.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43033/1/22_2005_Article_BF01226859.pd

Spin depolarization of muonium in mesoporous silica

We report muon spin rotation/relaxation measurements of muonium in mesoporous silica (SBA-15) with a high specific surface area of 600 m2/g. Up to 70 percent of the incoming muons form muonium and escape efficiently into the open pores at all temperatures between 3 and 300K. We present evidence that the interaction with the silica surfaces involves both spin exchange and a transition to a diamagnetic state, possibly due to dangling bonds on the surface. At very low temperatures, below 20K, the interaction between muonium and the silica surfaces is suppressed due to a He film coating the surfaces. These results indicate that it should be possible to use muonium to probe the surfaces of uncapped nanoparticles supported in silica

Harmonic Sums and Mellin Transforms up to two-loop Order

A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin transforms of all individual functions $f_i(x)$ of the momentum fraction $x$ emerging in the quantities of massless QED and QCD up to two--loop order, as the unpolarized and polarized splitting functions, coefficient functions, and hard scattering cross sections for space and time-like momentum transfer. The finite harmonic sums are calculated explicitly in the linear representation. Algebraic relations connecting these sums are derived to obtain representations based on a reduced set of basic functions. The Mellin transforms of all the corresponding Nielsen functions are calculated.Comment: 44 pages Latex, contract number adde

B-NMR of 8Li+ in rutile TiO2

We report preliminary low-energy B-NMR measurements of 8Li+ implanted in single crystal rutile TiO2 at an applied field of 6.55 T and 300 K. We observe a broad 12 kHz wide quadrupole split resonance with unresolved features and a sharp component at the Larmor frequency. The line broadening may be caused by overlapping multi-quantum transitions or motion of 8Li+ on the scale of its lifetime (1.21 s). We also find spin-lattice relaxation that is relatively fast compared to other wide band gap insulators. The origin of this fast relaxation is also likely quadrupolar and may be due to anisotropic 8Li+ diffusion