How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf \Sigma^0_2}$ sets. It follows that complementation for tree automata is provable from $\Pi^1_3$- but not $\Delta^1_3$-comprehension. We then use results due to MedSalem-Tanaka, M\"ollerfeld and Heinatsch-M\"ollerfeld to prove that over $\Pi^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf \Sigma^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $\Pi^1_3$-reflection principle for $\Pi^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $\Delta^1_3$-comprehension.Comment: 21 page

Sampling Distributions of Random Electromagnetic Fields in Mesoscopic or Dynamical Systems

We derive the sampling probability density function (pdf) of an ideal localized random electromagnetic field, its amplitude and intensity in an electromagnetic environment that is quasi-statically time-varying statistically homogeneous or static statistically inhomogeneous. The results allow for the estimation of field statistics and confidence intervals when a single spatial or temporal stochastic process produces randomization of the field. Results for both coherent and incoherent detection techniques are derived, for Cartesian, planar and full-vectorial fields. We show that the functional form of the sampling pdf depends on whether the random variable is dimensioned (e.g., the sampled electric field proper) or is expressed in dimensionless standardized or normalized form (e.g., the sampled electric field divided by its sampled standard deviation). For dimensioned quantities, the electric field, its amplitude and intensity exhibit different types of Bessel $K$ sampling pdfs, which differ significantly from the asymptotic Gauss normal and $\chi^{(2)}_{2p}$ ensemble pdfs when $\nu$ is relatively small. By contrast, for the corresponding standardized quantities, Student $t$, Fisher-Snedecor $F$ and root-$F$ sampling pdfs are obtained that exhibit heavier tails than comparable Bessel $K$ pdfs. Statistical uncertainties obtained from classical small-sample theory for dimensionless quantities are shown to be overestimated compared to dimensioned quantities. Differences in the sampling pdfs arising from de-normalization versus de-standardization are obtained.Comment: 12 pages, 15 figures, accepted for publication in Phys. Rev. E, minor typos correcte

A new class of semiclassical wave function uniformizations

We present a new semiclassical technique which relies on replacing complicated classical manifold structure with simpler manifolds, which are then evaluated by the usual semiclassical rules. Under circumstances where the original manifold structure gives poor or useless results semiclassically the replacement manifolds can yield remarkable accuracy. We give several working examples to illustrate the theory presented here.Comment: 12 pages (incl. 12 figures

New Complexity Results and Algorithms for the Minimum Tollbooth Problem

The inefficiency of the Wardrop equilibrium of nonatomic routing games can be eliminated by placing tolls on the edges of a network so that the socially optimal flow is induced as an equilibrium flow. A solution where the minimum number of edges are tolled may be preferable over others due to its ease of implementation in real networks. In this paper we consider the minimum tollbooth (MINTB) problem, which seeks social optimum inducing tolls with minimum support. We prove for single commodity networks with linear latencies that the problem is NP-hard to approximate within a factor of $1.1377$ through a reduction from the minimum vertex cover problem. Insights from network design motivate us to formulate a new variation of the problem where, in addition to placing tolls, it is allowed to remove unused edges by the social optimum. We prove that this new problem remains NP-hard even for single commodity networks with linear latencies, using a reduction from the partition problem. On the positive side, we give the first exact polynomial solution to the MINTB problem in an important class of graphs---series-parallel graphs. Our algorithm solves MINTB by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally

Atom cooling and trapping by disorder

We demonstrate the possibility of three-dimensional cooling of neutral atoms by illuminating them with two counterpropagating laser beams of mutually orthogonal linear polarization, where one of the lasers is a speckle field, i.e. a highly disordered but stationary coherent light field. This configuration gives rise to atom cooling in the transverse plane via a Sisyphus cooling mechanism similar to the one known in standard two-dimensional optical lattices formed by several plane laser waves. However, striking differences occur in the spatial diffusion coefficients as well as in local properties of the trapped atoms.Comment: 11 figures (postscript

Whirlpool: Improving Dynamic Cache Management with Static Data Classification

Cache hierarchies are increasingly non-uniform and difficult to manage. Several techniques, such as scratchpads or reuse hints, use static information about how programs access data to manage the memory hierarchy. Static techniques are effective on regular programs, but because they set fixed policies, they are vulnerable to changes in program behavior or available cache space. Instead, most systems rely on dynamic caching policies that adapt to observed program behavior. Unfortunately, dynamic policies spend significant resources trying to learn how programs use memory, and yet they often perform worse than a static policy. We present Whirlpool, a novel approach that combines static information with dynamic policies to reap the benefits of each. Whirlpool statically classifies data into pools based on how the program uses memory. Whirlpool then uses dynamic policies to tune the cache to each pool. Hence, rather than setting policies statically, Whirlpool uses static analysis to guide dynamic policies. We present both an API that lets programmers specify pools manually and a profiling tool that discovers pools automatically in unmodified binaries. We evaluate Whirlpool on a state-of-the-art NUCA cache. Whirlpool significantly outperforms prior approaches: on sequential programs, Whirlpool improves performance by up to 38% and reduces data movement energy by up to 53%; on parallel programs, Whirlpool improves performance by up to 67% and reduces data movement energy by up to 2.6x.National Science Foundation (U.S.) (grant CCF-1318384)National Science Foundation (U.S.) (CAREER-1452994)Samsung (Firm) (GRO award

CF3 Rotation in 3-(Trifluoromethyl)phenanthrene. X-ray Diffraction and ab Initio Electronic Structure Calculations

The molecular and crystal structure of 3-(trifluoromethyl)phenanthrene has been determined by X-ray diffraction. The structure of the isolated molecule has been calculated using electronic structure methods at the HF/3-21G, HF/6-31G*, MP2/6-31G* and B3LYP/6-31G* levels. The potential energy surfaces for the rotation of the CF3 group in both the isolated molecule and cluster models for the crystal were computed using electronic structure methods. The barrier height for CF3 rotation in the isolated molecule was calculated to be 0.40 kcal mol-1 at B3LYP/6-311+G**//B3LYP/6-311+G**. The B3LYP/6-31G* calculated CF3 rotational barrier in a 13-molecule cluster based on the X-ray data was found to be 2.6 kcal mol-1. The latter is in excellent agreement with experimental results from the NMR relaxation experiments reported in the companion paper (Beckmann, P. A.; Rosenberg, J.; Nordstrom, K.; Mallory, C. W.; Mallory, F. B. J. Phys. Chem. A 2006, 110, 3947). The computational results on the models for the solid state suggest that the intermolecular interaction between nearest neighbor pairs of CF3 groups in the crystal accounts for roughly 75% of the barrier to rotation in the solid state. This pair is found to undergo cooperative reorientation. We attribute the CF3 reorientational disorder in the crystal as observed by X-ray diffraction to the presence of a pair of minima on the potential energy surface and the effects of librational motion