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ON THE r-TH HYPER-KLOOSTERMAN SUMS AND ITS HYBRID MEAN VALUE
Zhang, Tianping,Zhang, Wenpeng Korean Mathematical Society 2006 대한수학회지 Vol.43 No.6
The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the T-th hyper-Kloosterman sums Kl(h, k+1, r;q) and the hyper Cochrane sums C(h, q; m, k), and give an interesting mean value formula.
On the $r$-th hyper-Kloosterman sums and its hybrid mean value
Tianping Zhang,Wenpeng Zhang 대한수학회 2006 대한수학회지 Vol.43 No.6
The main purpose of this paper is using the propertiesof Gauss sums, primitive characters and the mean value theoremsof Dirichlet L-functions to study the hybrid mean value of the r-thhyper-Kloosterman sums Kl (h;k+1 ;r;q) and the hyper Cochranesums C(h;q;m;k);and give an interesting mean value formula.
Hou, Wenpeng,Zhao, Ning-Jiu,Meng, Dongli,Tang, Jing,Zeng, Yi,Wu, Yu,Weng, Yangziwan,Cheng, Chungui,Xu, Xiulai,Li, Yi,Zhang, Jian-Ping,Huang, Yong,Bielawski, Christopher W.,Geng, Jianxin American Chemical Society 2016 ACS NANO Vol.10 No.5
<P>The installation of heterojunctions on the surfaces of carbon nanotubes (CNTs) is an effective method for promoting the charge separation processes needed for CNT-based electronics and optoelectronics applications. Conjugated polymers are proven state-of-the-art candidates for modifying the surfaces of CNTs. However, all previous attempts to incorporate conjugated polymers to CNTs resulted in unordered interfaces. Herein we show that well-defined chains of regioregular poly(3-hexylthiophene) (P3HT) were successfully grown from the surfaces of multiwalled CNTs (MWNTs) using surface-initiated Kumada catalyst-transfer polycondensation. The polymerization was found to proceed in a controlled manner as chains of tunable lengths were prepared through variation of the initial monomer-to-initiator ratio. Moreover, it was determined that large-diameter MWNTs afforded highly ordered P3HT aggregates, which exhibited a markedly bathochromically shifted optical absorption due to a high grafting density induced planarization of the polymer chains. Using ultrafast spectroscopy, the heterojunctions formed between the MWNTs and P3HT were shown to effectively overcome the binding energy of excitons, leading to photoinduced electron transfer from P3HT to MWNTs. Finally, when used as prototype devices, the individual MWNT-g-P3HT core-shell structures exhibited excellent photoresponses under a low illumination density.</P>
Hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums
Huaning Liu,Wenpeng Zhang 대한수학회 2007 대한수학회지 Vol.44 No.1
The main purpose of this paper is to use the properties ofprimitive characters, Gaus ums and Ramanujan’s um to stu dy thehybrid mean value of generalized Bernouli numbers, general Klostermansums and Gaus ums, and give two asymptotic formulae.
Li Deng,Wenpeng Jia,Wen Zheng,Hao Liu,Degang Jiang,Zimeng Li,Yu Tian,Wenling Zhang,Jingquan Liu 한국공업화학회 2018 Journal of Industrial and Engineering Chemistry Vol.58 No.-
Ni–Al layered double hydroxides (Ni–Al LDHs) are adopted as an electrorheological (ER) materialascribed to their special layered structure, ample polarizable groups and good dispersibility. Themagnetite nanoparticles (Fe3O4 NPs) are utilized as classical magnetorheological (MR) materials onaccount of their excellent paramagnetic property. In this work, we report the rational design of highperformanceER/MR materials by integrating the structural merits of Fe3O4 NPs onto a binary layeredstructure system. The obtained Fe3O4/Ni–Al LDH NCs dispersed in silicone oil displayed appealing dualER/MR performances. The work opens a new path toward novel stimuli-responsive materials forindustrial applications.
On the 2k-th power mean value of the generalized quadratic Gauss sums
Yanfeng He,Wenpeng Zhang 대한수학회 2011 대한수학회보 Vol.48 No.1
The main purpose of this paper is using the elementary and analytic methods to study the properties of the 2k-th power mean value of the generalized quadratic Gauss sums, and give two exact mean value formulae for k=3 and 4.
A NOTE ON THE TWISTED LERCH TYPE EULER ZETA FUNCTIONS
He, Yuan,Zhang, Wenpeng Korean Mathematical Society 2013 대한수학회보 Vol.50 No.2
In this note, the $q$-extension of the twisted Lerch Euler zeta functions considered by Jang [Bull. Korean Math. Soc. 47 (2010), no. 6, 1181-1188] is further investigated, and the generalized multiplication theorem for the $q$-extension of the twisted Lerch Euler zeta functions is given. As applications, some well-known results in the references are deduced as special cases.
ON THE 2k-TH POWER MEAN VALUE OF THE GENERALIZED QUADRATIC GAUSS SUMS
He, Yanfeng,Zhang, Wenpeng Korean Mathematical Society 2011 대한수학회보 Vol.48 No.1
The main purpose of this paper is using the elementary and analytic methods to study the properties of the 2k-th power mean value of the generalized quadratic Gauss sums, and give two exact mean value formulae for k = 3 and 4.
A new mean value related to D. H. Lehmer's problem and Kloosterman sums
Di Han,Wenpeng Zhang 대한수학회 2015 대한수학회보 Vol.52 No.1
Let $q>1$ be an odd integer and $c$ be a fixed integer with $(c, q)=1$. For each integer $a$ with $1\le a \leq q-1$, it is clear that there exists one and only one $b$ with $0\leq b \leq q-1$ such that $ab \equiv c $ (mod $q$). Let $N(c, q)$ denote the number of all solutions of the congruence equation $ab \equiv c$ (mod $q$) for $1 \le a, b \leq q-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1$ $(\bmod q)$. The main purpose of this paper is using the mean value theorem of Dirichlet $L$-functions to study the mean value properties of a summation involving $\left(N(c, q)-\frac{1}{2}\phi(q)\right)$ and Kloosterman sums, and give a sharper asymptotic formula for it.