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.     

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.     

容量法测定氧化钒中的各种价态钒

用NaOH溶液处理样品，分离V（V）然后以Fe(Ⅱ）滴定测定之。定量加入过量的V（V）到上述残渣中并用H2SO4溶解，用Fe(Ⅱ）滴定，计算出V（Ⅱ），V（Ⅲ）的含量。V（Ⅳ）的测定是通过从总钒量中扣除V（Ⅱ）、V（Ⅲ）、V（Ⅴ）的含量得到。V（Ⅱ）、V（Ⅲ）和V（Ⅴ）的回收率均大于96％，其8次测定结果的相对标准偏差分别为：9．8％、0．4％、1．2％；本法快速、简便，在实际应用中效果良好。     

有机锗化合物中碳氢的快速测定

报道了用Co3O4 作覆盖氧化剂分析有机锗化合物中碳氢元素的方法。通过对 9个样品的十几次分析 ,碳、氢的相对误差分别小于± 1.30 % ,4 .4 0 %。方法简便 ,结果稳定可靠 ,可用于含锗有机化合物的碳氢分析。     

固体磷酸催化剂的活性相

研究了固体磷酸催化剂活性与其自由磷酸聚合态及磷酸硅晶型的关系，指出高活性磷酸催化剂应具备的条件，即自由磷酸中应为正磷酸和焦磷酸的混合物，结合态磷应具有“Ｃ”晶型。     

C12-s-C12·2Br与低碳醇混合水溶液胶团化行为的温度效应和焓/熵补偿

季铵盐Gemini表面活性剂C12-s-C12·2Br(s=2,3,4,6)与丙醇、丁醇、戊醇、己醇混合水溶液的In(cmc)随温度升高而逐渐增大.计算所得热力学数据表明,C12-s-C12·2Br与醇混合胶团化过程服从熵驱动机理,也出现了焓/熵补偿现象.随着温度上升,熵驱动力增大,在指定温度时,醇分子烷烃链上碳原子数n增大使△Gm0值减小,胶团结构更加稳定;而增加s使值增大,胶团稳定性下降.     

脲醛树脂合成反应过程的FTIR研究

利用傅里叶变换红外(FTIR)光谱研究了脲醛树脂合成中不同结构形成和变化情况. 对FTIR谱图解析和分析的结果表明, 在脲醛树脂预聚物合成过程中, 随着甲醛与尿素反应的进行, 红外谱图特征峰发生有规律的变化, 酰胺II带特征峰波数逐渐低移, 有更多羟甲基、醚键和—NHCH2—结构基元形成, 固化反应使上述结构基元相应减少. 着重分析了不同甲醛与尿素摩尔比(nF/nU)下尿素与甲醛反应产物结构的变化, 随nF/nU的增加, 酰胺II带特征峰波数逐渐低移, 碱性条件下醚键和—NHCH2—结构基元明显增加, 在较强酸性条件下Uron环、亚甲基桥和羟甲基显著增加, 而—NHCH2—结构基元含量下降.     

Pt／钇稳定氧化锆固体电解质在高温下的电化学性质

用交流阻抗技术研究了二电极、三电极Ｐｔ／钇稳定氧化锆（简称ＹＳＺ）高温固体电化学体系．开路电位下，Ｐｔ／ＹＳＺ体系只有一个阻抗半圆，对应于电极体系的电化学活化控制过程，极化电阻随温度变化的表观活化能为１７１．５ｋＪ／ｍｏｌ．Ｐｔ／ＹＳＺ界面的双电层电容约为３００μＦ／ｃｍ２．阳极极化下，交流阻抗极化电阻显著减小；阴极极化下，极化电阻反而增大，并出现浓差控制现象．     

Selenium Nanoparticles Prepared from Reverse Microemulsion Process

Selenium nanoparticles were prepared by a reverse microemulsion system. Sodium selenosulfate was used as selenium source. The results showed that hydrochloric acid concentration and reaction temperature had great influence on the morphology of products. The crystalline selenium nanowires and amorphous selenium nanorods were obtained in given condition.     

D-型氨基酸的生物法制备

自然界存在20种氨基酸,均为L-型氨基酸,它们是组成蛋白质的基本结构单元.所以,人们一般认为D-氨基酸在自然界中极少存在,因此对于D-氨基酸的功能或性质方面的研究也非常少.近年来,随着科学研究的深入,人们发现非天然的D-型氨基酸虽然不是构成蛋白质的基本结构单元,但许多植物、