Fast compact prime number sieves (among others)

A parameterized family of algorithms is presented for the problem of finding all prime numbers up to a limit N. Many previous algorithms for this problem are shown to correspond to members of this family. A particular choice of parameters leads to an algorithm that requires only Θ(N) additions and that runs in bits. The small storage requirement is made possible by a provably compact method of storing all the primes up to some limit.     

A study of product ratios in the direct (313 nm) and Hg-sensitized (254 nm) photodecomposition of 2-methylcyclohexanone in the gas phase

The photolysis of 2-methylcyclohexanone has been studied in the gas phase at 313 nm, mainly at 100°C, over a range of pressures. The Hg(6³P₁) photosensitized decomposition has also been investigated at 100°C. Under conditions of high excitation and/or little collisional quenching the major products are carbon monoxide and the hydrocarbons: 1-hexene, trans-and cis-2-hexene and methylcyclopentane, with minor aldehyde formation. The various product ratios are presented in tabular form. At lower excitation energies, and with increased collisional deactivation, trans- and cis-5-heptenal become important products, and the trans/cis aldehyde ratio is seen to be slightly pressure dependent when all the systems are compared. Similarly, there is a small pressure dependence for the Σ hexenes/methylcyclopentane ratio. From experiments at 250°C the temperature dependence of this ratio was established, and for thermalized hexane-1,5-diyl an activation energy difference (E_d ? E_c) = ?1.3 kcal mol^?1 has been determined for the disproportionation and combination of the biradical. The mechanism for the photolysis is discussed in terms of triplet state photochemistry and biradical intermediates as developed, in particular, by Frey and coworkers, this Journal, 16 , 1337 (1984).     

Spectral interference on the emission of sulphur I 180.73 nm in an inductively coupled plasma

Spectral interferences on the S I 180.73 nm emission line by Al, Ca, Mn and Fe in an inductively coupled plasma are described. The critical concentration ratios for Ca, Mn, Al and Fe with respect to S are 90, 300, 3300 and 25000 respectively (full width at half maximum = 0.0075 nm). Intensity measurements at various observation zones in the plasma show that the spectral interference from Ca and Mn emission line overlaps may be reduced by observing the S emission at a lower observation height in the plasma.     

QCD calculations in the light-cone gauge

The non-singlet quark structure function is calculated in the leading logarithm approximation in an axial gauge with n² = 0, the light-cone gauge. The choice n² = 0 leads to a simple identity for loop integrals involving the extra n · k denominators. We compare the results graph by graph with both Feynman gauge QCD and a scalar gluon theory. The leading diagrams are the same “rainbow” diagrams as for the case of the scalar theory.The techniques are also applied to quark-quark scattering at large transverse momentum. The leading diagrams have the same dressed ladder-form factor structure.     

Isotopic mixing for the determination of relative coverages in overlayer structures: CO on Cu(111)

An accurate determination of the relative uptakes of CO in the formation of three successive overlayer structures on Cu(111) has been achieved by sequential adsorption of ¹²CO and ¹³CO followed by isotopic analysis of the desorbing gas. The results identify the intermediate structure as (1.5 × 1.5)?18° at a coverage of 0.44, rather than a slightly distorted c(4 × 2) structure with a coverage of 0.50.     

Integrality gap of the hypergraphic relaxation of Steiner trees: A short proof of a 1.55 upper bound

Recently, Byrka, Grandoni, Rothvoß and Sanità gave a 1.39 approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a shorter proof of the same integrality gap bound, by applying some of their techniques to a randomized loss-contracting algorithm.     

Calculation of molecular g-tensors using the zeroth-order regular approximation and density functional theory: expectation value versus linear response approaches

Density functional theory (DFT) calculations of molecular g-tensors were implemented as a second derivative property within the two-component relativistic zeroth-order regular approximation (ZORA). g-tensors were computed for systems ranging from light atomic radicals to molecules with heavy d and f block elements. For comparison, computations were also performed with a ZORA first-order derivative approach and with a second derivative method based on the Pauli Hamiltonian. In each set of computations, Slater-type basis sets have been used. The new ZORA implementation allows for non-hybrid and hybrid DFT calculations. A comparison of the PBE non-hybrid and the PBE0 hybrid functional yielded mixed results for our test set. For the lanthanide complex $[\hbox{Ce}(\hbox{DPA})_3]^{3-}$ (DPA = pyridine-2,6-dicarboxylate), calculations of the g-tensor were used to estimate paramagnetic NMR pseudocontact shifts for protons and carbon atoms in the ligands. The results are in reasonable agreement with experimental data.     

Bragg scattering as a probe of atomic wave functions and quantum phase transitions in optical lattices

We have observed Bragg scattering of photons from quantum degenerate ^{87}Rb atoms in a three-dimensional optical lattice. Bragg scattered light directly probes the microscopic crystal structure and atomic wave function whose position and momentum width is Heisenberg limited. The spatial coherence of the wave function leads to revivals in the Bragg scattered light due to the atomic Talbot effect. The decay of revivals across the superfluid to Mott insulator transition indicates the loss of superfluid coherence.