On a class of relatively prime sequences

AbstractFor each natural number n, let a0(n) = n, and if a0(n),…,ai(n) have already been defined, let ai+1(n) > ai(n) be minimal with (ai+1(n), a0(n) … ai(n)) = 1. Let g(n) be the largest ai(n) not a prime or the square of a prime. We show that g(n) ∼ n and that g(n) > n + cn12log(n) for some c > 0. The true order of magnitude of g(n) − n seems to be connected with the fine distribution of prime numbers. We also show that “most” ai(n) that are not primes or squares of primes are products of two distinct primes. A result of independent interest comes of one of our proofs: For every sufficiently large n there is a prime p < n12 with [np] composite

Modeling the observed tropospheric BrO background: Importance of multiphase chemistry and implications for ozone, OH, and mercury

Aircraft and satellite observations indicate the presence of ppt (ppt ≡ pmol/mol) levels of BrO in the free troposphere with important implications for the tropospheric budgets of ozone, OH, and mercury. We can reproduce these observations with the GEOS-Chem global tropospheric chemistry model by including a broader consideration of multiphase halogen (Br–Cl) chemistry than has been done in the past. Important reactions for regenerating BrO from its non-radical reservoirs include HOBr+Br−/Cl− in both aerosols and clouds, and oxidation of Br− by ClNO3 and ozone. Most tropospheric BrO in the model is in the free troposphere, consistent with observations, and originates mainly from the photolysis and oxidation of ocean-emitted CHBr3. Stratospheric input is also important in the upper troposphere. Including production of gas phase inorganic bromine from debromination of acidified sea salt aerosol increases free tropospheric Bry by about 30 %. We find HOBr to be the dominant gas-phase reservoir of inorganic bromine. Halogen (Br-Cl) radical chemistry as implemented here in GEOS-Chem drives 14 % and 11 % decreases in the global burdens of tropospheric ozone and OH, respectively, a 16 % increase in the atmospheric lifetime of methane, and an atmospheric lifetime of 6 months for elemental mercury. The dominant mechanism for the Br-Cl driven tropospheric ozone decrease is oxidation of NOx by formation and hydrolysis of BrNO3 and ClNO3

Angular Conditions,Relations between Breit and Light-Front Frames, and Subleading Power Corrections

We analyze the current matrix elements in the general collinear (Breit) frames and find the relation between the ordinary (or canonical) helicity amplitudes and the light-front helicity amplitudes. Using the conservation of angular momentum, we derive a general angular condition which should be satisfied by the light-front helicity amplitudes for any spin system. In addition, we obtain the light-front parity and time-reversal relations for the light-front helicity amplitudes. Applying these relations to the spin-1 form factor analysis, we note that the general angular condition relating the five helicity amplitudes is reduced to the usual angular condition relating the four helicity amplitudes due to the light-front time-reversal condition. We make some comments on the consequences of the angular condition for the analysis of the high-$Q^2$ deuteron electromagnetic form factors, and we further apply the general angular condition to the electromagnetic transition between spin-1/2 and spin-3/2 systems and find a relation useful for the analysis of the N-$\Delta$ transition form factors. We also discuss the scaling law and the subleading power corrections in the Breit and light-front frames.Comment: 24 pages,2 figure

Effect of stress level on the high temperature deformation and fracture mechanisms of Ti-45Al-2Nb-2Mn-0.8 vol. pct TiB²: an 'In Situ' experimental study

The effect of the applied stress on the deformation and crack nucleation and propagation mechanisms of a gamma-TiAl intermetallic alloy (Ti-45Al-2Nb-2Mn (at. pct)-0.8 vol. pct TiB2) was examined by means of in situ tensile (constant strain rate) and tensile-creep (constant load) experiments performed at 973 K (700 °C) using a scanning electron microscope. Colony boundary cracking developed during the secondary stage in creep tests at 300 and 400 MPa and during the tertiary stage of the creep tests performed at higher stresses. Colony boundary cracking was also observed in the constant strain rate tensile test. Interlamellar ledges were only found during the tensile-creep tests at high stresses (sigma > 400 MPa) and during the constant strain rate tensile test. Quantitative measurements of the nature of the crack propagation path along secondary cracks and along the primary crack indicated that colony boundaries were preferential sites for crack propagation under all the conditions investigated. The frequency of interlamellar cracking increased with stress, but this fracture mechanism was always of secondary importance. Translamellar cracking was only observed along the primary crack.Funding from the Spanish Ministry of Science and Innovation through projects (MAT2009-14547-C02-01 and MAT2009-14547-C02-02) is acknowledged. The Madrid Regional Government partially supported this project through the ESTRUMAT grant (P2009/MAT-1585). CJB acknowledges the support from the Spanish Ministry of Education for his sabbatical stay in Madrid (SAB2009-0045).Publicad

In Situ Observations of the Deformation Behavior and Fracture Mechanisms of Ti-45Al-2Nb-2Mn+0.8 vol pct TiB₂

The deformation and fracture mechanisms of a nearly lamellar Ti-45Al-2Nb-2Mn (at. pct) + 0.8 vol pct TiB₂ intermetallic, processed into an actual low-pressure turbine blade, were examined by means of in situ tensile and tensile-creep experiments performed inside a scanning electron microscope (SEM). Low elongation-to-failure and brittle fracture were observed at room temperature, while the larger elongations-to-failure at high temperature facilitated the observation of the onset and propagation of damage. It was found that the dominant damage mechanisms at high temperature depended on the applied stress level. Interlamellar cracking was observed only above 390 MPa, which suggests that there is a threshold below which this mechanism is inhibited. Failure during creep tests at 250 MPa was controlled by intercolony cracking. The in situ observations demonstrated that the colony boundaries are damage nucleation and propagation sites during tensile creep, and they seem to be the weakest link in the microstructure for the tertiary creep stage. Therefore, it is proposed that interlamellar areas are critical zones for fracture at higher stresses, whereas lower stress, high-temperature creep conditions lead to intercolony cracking and fracture.The authors are grateful to Industria de Turbo Propulsores, S.A. for supplying the intermetallic blades. Funding from the Spanish Ministry of Science and Innovation through projects MAT2009-14547-C02-01 and MAT2009-14547-C02-02 is acknowledged. The Madrid Regional Government supported this project partially through the ESTRUMAT grant P2009/MAT-1585. C.J.B. acknowledges the support from Grant SAB2009-0045 from the Spanish Ministry of Education for his sabbatical stage in Madrid.Publicad