Extremal Cubic Inequalities of Three Variables

Let $\mathcal{H}_{3,d}$ be the vector space of homogeneous three variable polynomials of degree $d$, and $\mathcal{P}_{3,d}^+$ be the set of all elements $f \in \mathcal{H}_{3,d}$ such that$f(x,y,z) \geq 0$ for all $x \geq 0$, $y \geq 0$, $z \geq 0$. In this article, we determine all extremal elements of $\mathcal{P}_{3,3}^+$. We prove that if $f \in \mathcal{P}_{3,3}^+$ is an irreducible extremal element, then the zero locus $V_{\mathbb{C}}(f)$ in $\mathbb{P}_{\mathbb{C}}^2$ is a rational curve whose singularity is an acnode in the interior of $\mathbb{P}_+^2$ or a cusp on an edge of $\mathbb{P}_+^2$. We also prove that if $f \in \mathcal{P}_{3,3}^+$ is an extremal element, then $f(x^2,y^2,z^2)$ is an extremal element of $\mathcal{P}_{3,6}$, where $\mathcal{P}_{3,d}$ is the set of all the elements $f \in \mathcal{H}_{3,d}$ such that $f(x,y,z) \geq 0$ for all $x$, $y$, $z \in \mathbb{R}$. A notion of infinitely near zeros of an inequality is introduced, and plays an important role

Ambipolar transport in bulk crystals of a topological insulator by gating with ionic liquid

We report that the ionic-liquid gating of bulk single crystals of a topological insulator can control the type of the surface carriers and even results in ambipolar transport. This was made possible by the use of a highly bulk-insulating BiSbTeSe2 system where the chemical potential is located close to both the surface Dirac point and the middle of the bulk band gap. Thanks to the use of ionic liquid, the control of the surface chemical potential by gating was possible on the whole surface of a bulk three-dimensional sample, opening new experimental opportunities for topological insulators. In addition, our data suggest the existence of a nearly reversible electrochemical reaction that causes bulk carrier doping into the crystal during the ionic-liquid gating process.Comment: 7 pages, 6 figures, 2 tables; significantly expanded version to fully discuss the gating process and its side effects; published in PR

Effect of Specimen Size on Toughness Evaluation by Charpy Test

Herein we present an experimental investigation on the toughness evaluation method for the samples of copper and aluminum, which are generally employed as electronic equipment parts, through the miniaturized version of the Charpy impact test. Overall, the resulting high reproducibility of the absorbed energy values informed by the miniaturized Charpy impact test can be witnessed; moreover, it is possible to compare the values given by the Japanese industrial standard (JIS) Charpy impact test to those given by the miniaturized Charpy impact test and correction factors were calculated accordingly

Effect of Rust Inhibitor in Brine on Corrosion Properties of Copper

In this study, the effects of rust inhibitors in brine on corrosion behaviors of copper were investigated by measurement of cathode and anode polarization curves and an immersion test. For rust inhibitors, benzotriazole, sodium benzoate and sodium nitrite were prepared. From measurement results of cathode and anode polarization curves, it was found that the corrosion rate of copper in the benzotriazole solution is low and a stable passive film with excellent corrosion resistance generates on the surface of copper in the solution. In the case of the sodium benzoate solution, the corrosion resistance of the passive film was inferior to that in the benzotriazole solution although the passive film generated on the surface of copper. In contrast, the passive film scarcely generated on the surface of copper in the sodium nitrite solution. The result shows that the rust preventive effect of the solution to copper is weak. Furthermore, the immersion test revealed that the benzotriazole solution has the rust preventive effect to copper. In contrast, the effect of the sodium benzoate solution is weak and that of the sodium nitrite solution is scarcely expected