Maximum Gap in (Inverse) Cyclotomic Polynomial

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(\Phi_n)$ and $g(\Psi_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(\Phi_{p_1})$, $g(\Psi_{p_1})$ and $g(\Psi_{p_1p_2})$. Hence the simplest non-trivial cases are $g(\Phi_{p_1p_2})$ and $g(\Psi_{p_1p_2p_3})$. We provide an exact expression for $g(\Phi_{p_1p_2}).$ We also provide an exact expression for $g(\Psi_{p_1p_2p_3})$ under a mild condition. The condition is almost always satisfied (only finite exceptions for each $p_1$). We also provide a lower bound and an upper bound for $g(\Psi_{p_1p_2p_3})$

Semiconductor integrated circuit chip, multilayer chip capacitor and semiconductor integrated circuit chip package

Disclosed are a semiconductor integrated circuit chip, a multilayer chip capacitor, and a semiconductor integrated circuit chip package. The semiconductor integrated circuit chip includes a semiconductor integrated circuit chip body, an input/output terminal disposed on the outside of the semiconductor integrated circuit chip body, and a decoupling capacitor disposed at a side face of the semiconductor integrated circuit chip body and electrically connected to the input/output terminal. The semiconductor integrated circuit chip cab be obtained, which can maintain an impedance of a power distribution network below a target impedance in a wide frequency range, particularly at a high frequency, by minimizing an inductance between a decoupling capacitor and a semiconductor integrated circuit chip

Broad Phase Transition of Fluorite-Structured Ferroelectrics for Large Electrocaloric Effect

Field-induced ferroelectricity in (doped) hafnia and zirconia has attracted increasing interest in energy-related applications, including energy harvesting and solid-state cooling. It shows a larger isothermal entropy change in a much wider temperature range compared with those of other promising candidates. The field-induced phase transition occurs in an extremely wide temperature range, which contributes to the giant electrocaloric effect. This article examines the possible origins of a large isothermal entropy change, which can be related to the extremely broad phase transitions in fluorite-structured ferroelectrics. While the materials possess a high entropy change associated with the polar–nonpolar phase transition, which can contribute to the high energy performance, the higher breakdown field compared with perovskites practically determines the available temperature range

Effect of cytokinins on growth and phenylpropanoid accumulation in Tartary buckwheat sprouts (Fagopyrum esculentum Moench)

This study analyzed the effect of plant hormones, zeatin, 6-benzyl amino purine (BAP), kinetin, and thidiazuron (TDZ) on the growth of Tartary buckwheat sprouts and analyzed the fresh weight, shoot and root length, and production of phenolic compounds. All the hormone-treated plants at the lowest concentration (0.1 mg/L) showed the highest levels of growth parameters (fresh weight, shoot, and root length) when compared to the control. Among the various hormones treatment, the plant treated with 1 mg/L of BAP, kinetin, and zeatin showed the highest total phenolic level, whereas the TDZ showed the highest accumulation of total phenolic at the lowest concentration (0.1 mg/L). A total of 6 compounds were identified (4-hydroxybenzoic acid, caffeic acid, chlorogenic acid, p-coumaric acid, rutin, and trans-cinnamic acid) were quantified by high liquid performance chromatography (HPLC) after treatment of plant with different concentrations of hormones. Among these individual phenolic compounds, at the higher hormonal concentration (1 mg/L) the rutin showed the highest accumulation in BAP, zeatin, and kinetin treated sprout, whereas in the TDZ treated sprout the rutin content was highest at the lowest concentration (0.1 mg/L). From these results, it is suggested that BAP, zeatin, and kinetin at the lowest concentrations might positively enhance the growth of buckwheat sprouts, whereas at the highest hormonal treatment the accumulation of the phenolic compounds was higher. However, in TDZ treatment the growth and phenolic compound accumulation were highest at the lowest concentration. From these results, it is showed that suitable concentrations might enhance the growth and phenolic compound accumulation in Tatary buckwheat sprout