This paper concerns the $L^4$ norm of Littlewood polynomials on the unit circle which are given by

$$ \begin{align*}q_n(z)=\sum_{k=0}^{n-1}\pm z^k;\end{align*} $$

$\{-1,1\}$

$$ \begin{align*}||q_n||_4^4=\frac{1}{2\pi}\int_0^{2\pi}|q_n(e^{i\theta})|^4 d\theta.\end{align*} $$

$||q_n||_4/\sqrt {n}\rightarrow \sqrt [4]{2}$

$n\to \infty $

Let $f(z)=\sum _{n=0}^{\infty }a_n z^n \in H(\mathbb {D})$ be an analytic function over the unit disk in the complex plane, and let $\mathcal {R} f$ be its randomization:

$$ \begin{align*}(\mathcal{R} f)(z)= \sum_{n=0}^{\infty} a_n X_n z^n \in H(\mathbb{D}),\end{align*} $$

where $(X_n)_{n\ge 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this note, we characterize those $f(z) \in H(\mathbb {D})$ such that the zero set of $\mathcal {R} f$ satisfies a Blaschke-type condition almost surely:

$$ \begin{align*}\sum_{n=1}^{\infty}(1-|z_n|)^t<\infty, \quad t>1.\end{align*} $$

Rapid infant growth increases the risk for adult obesity. The gut microbiome is associated with early weight status; however, no study has examined how interactions between microbial and host ribonucleic acid (RNA) expression influence infant growth. We hypothesized that dynamics in infant stool micro-ribonucleic acids (miRNAs) would be associated with both microbial activity and infant growth via putative metabolic targets. Stool was collected twice from 30 full-term infants, at 1 month and again between 6 and 12 months. Stool RNA were measured with high-throughput sequencing and aligned to human and microbial databases. Infant growth was measured by weight-for-length z-score at birth and 12 months. Increased RNA transcriptional activity of Clostridia (R = 0.55; Adj p = 3.7E-2) and Burkholderia (R = −0.820, Adj p = 2.62E-3) were associated with infant growth. Of the 25 human RNAs associated with growth, 16 were miRNAs. The miRNAs demonstrated significant target enrichment (Adj p < 0.05) for four metabolic pathways. There were four associations between growth-related miRNAs and growth-related phyla. We have shown that longitudinal trends in gut microbiota activity and human miRNA levels are associated with infant growth and the metabolic targets of miRNAs suggest these molecules may regulate the biosynthetic landscape of the gut and influence microbial activity.

The hydrogen concentration and composition of garnets in the ultrahigh pressure eclogites at Shuanghe, eastern Dabieshan, were investigated using Fourier transform infrared spectroscopy and electron microprobe analysis. The OH absorption bands can be divided into four groups: (1) 3635–3655 cm–1; (2) 3600–3630 cm–1; (3) 3540–3580 cm–1; and (4) 3400–3450 cm–1 and the water content ranges from 45 to 2529 ppm. Based on the behaviour of the OH absorption band and the relationship between water content and the composition of garnets, the samples can be divided into two classes: samples with >400 ppm H2O and samples with ≤400 ppm H2O. The water content of the former shows an obvious positive correlation with Ca atoms and a negative correlation with the Si, Mg and Fe2+ atoms per 12 anions, whereas the water content of the latter shows no obvious linear correlation with cations. It is concluded that the major mechanism of hydroxyl incorporation in garnets with >400 ppm H2O is by the coupled substitution 4H +Z□ → □+ZSi in the tetrahedral site, and that several mechanisms are responsible for OH incorporation in garnets with ≤400 ppm H2O.

This work designed a facile preparation for an SiO2/C composite as the anode material for lithium ion battery. Both SiO2 and carbon are amorphous. SiO2 and carbon are mixed uniformly. The SiO2/C composite shows high specific capacity, cycle stability, and rate capability in lithium ion battery charge–discharge test. A stable reversible capacity of 1024 mA h/g at the current density of 100 mA/g is reached. The capacity retains 83% after 100 cycles. The uniform mixture of SiO2 and carbon leads to reduced volume change during the lithiation and delithiation of SiO2, together with the amorphous nature of SiO2 explains the high cycling stability. The carbon coating is a key factor for the high capacity and stability due to the increased electrical conductivity and reduced volume change. The resistance of the solid electrolyte interface film and charge transfer resistance of the SiO2/C composite are much smaller than those of pure carbon, which is a direct proof of the improved conductivity of the material by the carbon coating.

The effect of Li3N additive on the Li-Mg-N-H system was examined with respect to the reversible dehydrogenation performance. Screening study with varying Li3N additions (5, 10, 20, and 30 mol%) demonstrates that all are effective for improving the hydrogen desorption capacity. Optimally, incorporation of 10 mol% Li3N improves the practical capacity from 3.9 wt% to approximately 4.7 wt% hydrogen at 200 °C, which drives the dehydrogenation reaction toward completion. Moreover, the capacity enhancement persists well over 10 de-/rehydrogenation cycles. Systematic x-ray diffraction examinations indicate that Li3N additive transforms into LiNH2 and LiH phases and remains during hydrogen cycling. Combined structure/property investigations suggest that the LiNH2 “seeding” should be responsible for the capacity enhancement, which reduces the kinetic barrier associated with the nucleation of intermediate LiNH2. In addition, the concurrent incorporation of LiH is effective for mitigating the ammonia release.