Explicit representations of the integral containing the error term in the divisor problem

We shall investigate several properties of the integral $$ \int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt} $$ with a natural number k, a non-negative integer j and a complex variable θ, where Δ _k (x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.     

Upper Bound on the Diameter of a Domination Dot-Critical Graph

A vertex of a graph is called critical if its deletion decreases the domination number, and an edge is called dot-critical if its contraction decreases the domination number. A graph is said to be dot-critical if all of its edges are dot-critical. In this paper, we show that if G is a connected dot-critical graph with domination number k??? 3 and diameter d and if G has no critical vertices, then d??? 2k?3.     

Measurement of sugars using the laser spray technique with a gold capillary

A gold (Au) capillary has higher thermal conductivity than a stainless steel capillary and can withstand capillary over-heating induced by high CO(2) laser irradiation (over 2.5 W) better than a stainless steel capillary. For this study, a laser spray using an Au capillary was applied for the detection of sugars. The signal of cationized compounds [M+Na](+) can be detected with higher sensitivity than with conventional laser sprays using high laser power (over 2.7 W). Using 3.5 W of laser power, the signal intensity is 15 times higher than the maximum value with stainless steel (2.3 W) in a 10(-5) M maltose aqueous solution. It is considered that almost all the water molecules evaporate by laser irradiation, which is impossible to achieve using a stainless steel capillary.     

Neutron Pairing Correlations in an {\alpha}–{n}–{n} Three-Cluster Model of the {^{6}}He Nucleus

An \({\alpha}\)–n–n three-cluster model of the \({^6}\)He nucleus is studied by solving the Faddeev equations, where the cluster potential between \({\alpha}\) and n takes into account the Pauli exclusion correction, using the Fish-Bone Optical Model (Schmid in Z Phys A 297:105, 1980). The resulting binding energy of the ground state (\({0^+}\)) is 0.831 MeV and the resonance energy of the first excited state (\({2^+}\)), 0.60–i0.012 MeV, is extracted from the three-cluster break-up threshold. These theoretical values are in reasonable agreement with the experimental data: 0.973 MeV and 0.824–i0.056 MeV, respectively. In order to investigate the structure of these states, we calculate the angle density matrix for the \({\angle n_1 \alpha n_2}\) angle in the triangle formed by the three clusters. The angle density matrix of the ground state has two peaks and the configuration of \({0^+}\) wave function corresponding to the peaks constitutes a mixture of an acute-angled triangle structure and an obtuse-angled one. This finding is consistent with the former result from a variational approach (Hagino and Sagawa in Phys Rev C 72:044321, 2005). On the other hand, in the case of \({2^+}\) state only a single peak is obtained.     

Infrared photodissociation spectroscopy of Al(+)(CH(3)OH)(n) (n = 1-4)

Infrared photodissociation spectra of Al(+)(CH(3)OH)(n) (n = 1-4) and Al(+)(CH(3)OH)(n)-Ar (n = 1-3) were measured in the OH stretching region, 3000-3800 cm(-1). For n = 1 and 2, sharp absorption bands were observed in the free OH stretching region, all of which were well reproduced by the spectra calculated for the solvated-type geometry with no hydrogen bond. For n = 3 and 4, there were broad vibrational bands in the energy region of hydrogen-bonded OH stretching vibrations, 3000-3500 cm(-1). Energies of possible isomers for the Al(+)(CH(3)OH)(3),4 ions with hydrogen bonds were calculated in order to assign these bands. It was found that the third and fourth methanol molecules form hydrogen bonds with methanol molecules in the first solvation shell, rather than a direct bonding with the Al(+) ion. For the Al(+)(CH(3)OH)(n) clusters with n = 1-4, we obtained no evidence of the insertion reaction, which occurs in Al(+)(H(2)O)(n). One possible explanation of the difference between these two systems is that the potential energy barriers between the solvated and inserted isomers in the Al(+)(CH(3)OH)(n) system is too high to form the inserted-type isomers.     

Total synthesis of borrelidin

The total synthesis of borrelidin has been achieved. The best feature of our synthetic route is macrocyclization at C11-C12 for the construction of an 18-membered ring after esterification between two segments. A detailed examination of the macrocyclization led us to the samarium(II) iodide-mediated intramolecular Reformatsky-type reaction as the most efficient synthetic approach. The two key segments were synthesized through regioselective methylation, directed hydrogenation, stereoselective Reformatsky-type reaction, and MgBr2.Et2O-mediated chelation-controlled allylation.     

Surfactant-mixing effects on the interfacial tension and the microemulsion formation in water/supercritical CO2 system

The effects of surfactant mixing on interfacial tension and on microemulsion formation were examined for systems of air/water and water/supercritical CO2 (scCO2) interfaces and for water/scCO2 microemulsions. A fluorinated surfactant, sodium bis(1H,1H,2H,2H-heptadecafluorodecyl)-2-sulfosuccinate (8FS(EO)2), was mixed with the three hydrocarbon surfactants, Pluronic L31, Tergitol TMN-6, and decyltrimethylammonium chloride (DeTAC), at equimolar ratio. For all the cases, the interfacial tension was significantly lowered by the mixing. The positive synergistic effect suggests that the mixed surfactants tend to pack more closely on the interface than the pure constituents. It was found, however, that the microemulsion formation in scCO2 was never facilitated by the mixing, except for the case of TMN-6. This is probably due to the segregation of the surfactants into hydrocarbon-rich and fluorocarbon-rich phases on the microemulsion surface.     

Estimation of the Refractive Index Change in Glass Induced by Femtosecond Laser Pulses

It is known that local refractive index change occurs when femtosecond laser pulses with extremely high peak power are launched into glass. We focused 130-femtosecond laser pulses of 800 nm into the bulk of glass and examined the shape of the induced refractive index change. We found that the length of the spot of the refractive index change along the optical axis reached about 30 μm despite the diameter being about 2 μm. To estimate the distribution of induced refractive index change, we fabricated Bragg grating by scanning the focused spot and calculated the amount of the change by applying Kogelnik’s coupled mode theory to the measured diffraction efficiencies of the higher order diffracted beams.