Arithmetic functions and the Cauchy product

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$nge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$nge 3$$).     

Biosynthesis of Indole Diterpene Lolitrems: Radical-Induced Cyclization of an Epoxyalcohol Affording a Characteristic Lolitremane Skeleton

Lolitrems are tremorgenic indole diterpenes that exhibit a unique 5/6 bicyclic system of the indole moiety. Although genetic analysis has indicated that the prenyltransferase LtmE and the cytochrome P450 LtmJ are involved in the construction of this unique structure, the detailed mechanism remains to be elucidated. Herein, we report the reconstitution of the biosynthetic pathway for lolitrems employing a recently established genome-editing technique for the expression host Aspergillus oryzae. Heterologous expression and bioconversion of the various intermediates revealed that LtmJ catalyzes multistep oxidation to furnish the lolitrem core. We also isolated the key reaction intermediate with an epoxyalcohol moiety. This observation allowed us to establish the mechanism of radical-induced cyclization, which was firmly supported by density functional theory calculations and a model experiment with a synthetic analogue.     

Generalized Cheeger-Gromoll metrics and the Hopf map

We show that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle, equipped with the induced metric, onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented.     

Fuzzy network technique for technological forecasting

The stochastic network technique is known to be a powerful tool carrying out a technological forecast of complex systems. A network dealt with is characterized by a tetrad of essential elements: logical nodes with some inputs and outputs, probabilistics activity branches, feedback loops, and multiple sources and sinks. A set of network parameters is defined for each element and their values are estimated for practical analysis of the network. In the case where the system to be treated is very large and/or complex, it cannot always be represented by a definite network and therefore forecasted values of parameters are inevitably indefinite themselves. A conventional probabilistic approach is sometimes inadequate in such a case. In the light of these facts, the paper proposes a fuzzy network technique, in which among activity branches emanating from a node, a branch to be undertaken once the node is realized belongs to a fuzzy set; and the time required to complete an activity branch belongs to a fuzzy set. Operations of maximum and minimum for sum and product of fuzzy sets take the place of manipulations of addition and multiplication for probabilities, respectively. Although the operations are somewhat formal, the obtained results seem interesting. A numerical example is attached to show a comparison of the proposed technique with the conventional one.     

Nanoparticles in Emissions and Atmospheric Environment: Now and Future

Journal of Nanoparticle Research -     

Magneto-Optical Measurement of Organic Conductor Using Rotational Cavity System

We have developed a new magneto-optical measurement system using a rotational cavity system equipped with a millimeter vector network analyzer and a 14 T solenoid type super conducting magnet. The measurement can be performed in the transmission configuration down to 1.6 K. The results of the precise angular dependence measurement of quasi-one-dimensional organic conductor (DMET)₂I₃ using the new system are shown, and its Fermi surface will be discussed in connection with the previous reports.     

Sound modes in holographic hydrodynamics for charged AdS black hole

In the previous paper we studied the transport coefficients of quark–gluon plasma in finite temperature and finite density in vector and tensor modes. In this paper, we extend it to the scalar modes. We work out the decoupling problem and hydrodynamic analysis for the sound mode in charged AdS black hole and calculate the sound velocity, the charge susceptibility and the electrical conductivity. We find that Einstein relation among the conductivity, the diffusion constant and the susceptibility holds exactly.     

Polarization-maintaining photonic crystal fibers with near-zero flattened dispersion in 1.06 μm waveband for medical applications

In this paper we present the design of a modified hexagonal photonic crystal fiber (PCF) having high birefringence and a near-zero flattened dispersion. Using the finite-difference method (FDM), it is shown that the proposed multiple Gedoped core hexagonal PCF exhibits a high birefringence of order 10⁻³ and a nearly zero flattened dispersion in the optical coherence tomography (OCT) waveband. In addition, the proposed PCF has a confinement loss of less than 10⁻⁸ dB/m at 1.06 μm. PCFs with such properties are considered suitable for both endoscopic OCT and other experimental setups employing 1.06 μm lasers.     

Pressure generation to 50 GPa in Kawai-type multianvil apparatus using newly developed tungsten carbide anvils

A pressure generation test for Kawai-type multianvil apparatus (KMA) has been made using second-stage anvils of a newly developed ultra-hard tungsten carbide composite. Superb performance of the new anvil with significantly less plastic deformation was confirmed as compared to those commonly used for the KMA experiments. A maximum pressure of ～48?GPa was achieved using the new anvils with a truncation edge length (TEL) of 1.5?mm, based on in situ X-ray diffraction measurements. Further optimization of materials and sizes of the pressure medium/gasket should lead to pressures even higher than 50?GPa in KMA using this novel tungsten carbide composite, which may also be used for expansion of the pressure ranges in other types of high pressure apparatus operated in large volume press.