The embedding theorem in limiting cases and its applications in singular perturbation

In 1980, Brézis^[6], using the technique of dividing the total space into two parts, proved the embedding theorem of limiting case which is very important in applications. In 1982, Ding Xiaqi improved the proof given in [6], by using of the technique of dividing the total space into three parts. In this paper, using the technique of dividing the total space into three parts, the author proves uniformly the results obtained by Ding^[3,4], and gives an embedding theorem of limiting case including (Lemma 2.2). And he also gives two kinds of examples, applying the embedding theorems (limiting case and non-limiting case) and the interpolation theorems. These examples are the singular perturbation problems in the sense of Lions^[1] (for the definition of singular perturbation, see [1], Introduction). But the singular solutionU _e converges uniformly to the limit solution (degenerate)U, ase0.     

On the maximum modulus of complete trigonometric sums

LetQ _k (p) be a set consisting of all polynomials of degreek with integral coefficientsf(x)=a _k x ^k +...+a ₁ x, wherep×a _k . For givenk andp any polynomialf _k,p (x)εQ _k (p) satisfying ‖S(p, f _k,p )‖=sup ‖S(p, f)‖fεQ(p) is called a maximum modular polynomial inQ _k (p), where $$S(p,f) = \sum\limits_{x = 0}^{p - 1} {e^{2\pi if(x)/p} } $$ Moreover, we definec(k, p)=‖S(p, f _k.p (x))‖. The main results are the following theorems.

1. For k=p?1 and p≥3 we have $$c(k,p) = \sqrt {p^2 - 4(p - 1)\sin ^2 \frac{\pi }{p}} $$ Besides, we may take \(f_{k,p} (x) = \prod\limits_{r = 0}^{p - 2} {(x - r)} \)
2. For k=p?s, 2≤s≤(p+1)/2 and p≥5, we have $$c(k,p) \leqslant p - 4(s - 1)\sin ^2 \frac{\pi }{p}$$ .

In Theorems 3 and 4, an interesting connextion between the present question and the famous problem of Prouhet and Tarry is given, some conditions under which the sign of equality in Theorem 2 holds are given and a method used to construct a maximum modular polynomial inQ _k (p) is also given.     

Carbon nanotube assisted synthesis of CeO2 nanotubes

CeO₂ nanotubes have been synthesized facilely using carbon nanotubes (CNTs) as templates by a liquid phase deposition method. The properties of the CeO₂ nanotubes were characterized by scanning electron microscopy (SEM), transmission electron microscopy (TEM), X-ray diffraction (XRD), X-ray photoelectron spectrum (XPS) as well as thermogravimetry and differential thermal analysis (TG-DTA). The obtained CeO₂ nanotubes with a polycrystalline face-centered cubic phase have a uniform diameter ranging from 40 to 50 nm. The CeO₂ nanotubes are composed of many tiny interconnected nanocrystallites of about 10 nm in size. The pretreatment of CNTs and calcination temperature were confirmed to be the crucial factors determining the formation of CeO₂ nanotubes. A possible formation mechanism has been suggested to explain the formation of CeO₂ nanotubes.     

All-atom molecular dynamic simulations and relative NMR spectra study of weak C-H...O contacts in amide-water systems

Amide-water mixtures are studied by all-atom molecular dynamics (MD) simulations and the relative temperature-dependent NMR experiment. The weak C-H...O contacts are found in the amide-water systems theoretically and experimentally. The statistical results of the average numbers of hydrogen bonds indicate that the methyl groups in amide molecules represent different capabilities in forming the weak C-H...O contacts. The statistics also imply that the C-H...O contacts are more obvious in the amide-rich region than those in the water-rich region. The temperature-dependent NMR spectra are also adopted to investigate the weak C-H...O contacts in the amide-water systems. The relative chemical shifts of the methyl groups are in good agreement with the MD simulations.