Existence of positive periodic solutions for two kinds of neutral functional differential equations

In this work, we deal with a new existence theory for positive periodic solutions for two kinds of neutral functional differential equations by employing the Krasnoselskii fixed-point theorem. Applying our results to various mathematical models we improve some previous results.     

Oxidation of lean formaldehyde in air over an Au/CeO2 catalyst and its kinetics

The oxidation of low concentration formaldehyde in air over Au/CeO₂, prepared by co-precipitation, was investigated. Power-law kinetic models were proposed and the parameters were estimated, which are r = −0.46 × e ^{−14612 / RT}C_HCHO(303 K < T < 363 K) and r = −295.78 × e ^{−34178 / RT} C_HCHO (363 K < T < 413 K). The mechanism of the reaction at low temperatures might be different from that at high temperatures.     

Existence of three positive solutions for some second-order m-point boundary value problems

By using fixed-point theorems, some new results for multiplicity of positive solutions for some second order m-point boundary value problems are obtained.The associated Green＇s function of these problems are also given.     

Morphology evolution of nanocomposites based on poly(phenylene sulfide)/poly(butylene terephthalate) blend

Poly(phenylene sulfide) (PPS)/poly(butylene terephthalate) (PBT) (60/40 w/w) blend nanocomposites (PPS/PBTs) were prepared by direct melt compounding of PPS, PBT, and organoclay. The morphology and rheology of PPS/PBTs were investigated using scanning electron microscope and transmission electron microscope as well as parallel plate rheometer. The intercalated clay tactoids are selectively located in the continuous PBT phase due to their nice affinity. A novel morphology evolution of the immiscible blend matrices is observed with increase of clay loadings. Small addition of clay increases the discrete PPS spherulite domain size. With increasing loading levels, the PPS phase transform to the fibrous structure and finally, to the partial laminar structure at the high loading levels, in which shows a characteristic of large‐scaled phase separation. The presence of clay, however, does not impede the coalescence of the PPS phase because the phase size increases with increasing clay loadings. The elasticity and blend ratio of two matrices are proposed as the important roles on the morphological evolution. Moreover, the laminar structure of PPS phase is very sensitive to the steady shear flow and is easy to be broken down to spherulite droplet at the low shear rate. However, high shear level is likely to facilitate the coalescence of those PPS phase and finally to phase inversion, both contributing to increases of the dynamic modulus after steady shear flow. In conclusion, the morphology of the immiscible polymer blend nanocomposites depends strongly on both the clay loadings and shear history. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1265–1279, 2008     

ON THE DIOPHANTINE EQUATION $\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$

In this paper,, the author proves the following result: Let $\[{E_{a,k}}(N)\]$ denote the number of natural numbers $\[n \le N\]$ for which equation $$\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$$ is insolable in positive integers $\[{x_i}(i = 0,1, \cdots ,k)\]$.Then $$\[{E_{a,k}}(N) \ll N\exp \{ - C{(\log N)^{1 - \frac{1}{{k + 1}}}}\} \]$$ where the implied constant depends on a and K.     

《数书九章》大衍类算题中的数论命题

《数书九章》是我国著名数学家南宋秦九韶撰写(公元1247年)。青年时他“侍亲中都(临安,今杭州市)因得访习于太史(国家天文历法官员)”。《数书九章》有“大衍总数术”以处理天文历法问题,系统论述同余式解法,这在我国历史上是第一次,在世界数学史上也是创举。 本文用整数论知识来检验《数书九章》中有关解同余式算题(大衍类)题文、术文、草文以及清代学者所作注文的得失,使对我国传统数学在这一领域内的工作有,较系统的认识。     

基于表观传递函数的象质评价及计算(Ⅱ)

首先讨论了表观传递函数的近似计算,然后用e指数近似公式进行了象差平衡并求得容限,为利用表观传递函数评价部分相干光学系统的成象质量提供了方法和依据。     

Studies on the structure-activity relationship of delta-sleep inducing peptide

DSIP and its fourteen analogs as well as three short peptides were synthesized by solid phase method. The design of the analogs was based on the consideration of the introduction of D-amino acid into the molecules to inhibit the enzymatic hydrolysis and the introduction of amino acids with a hydrophobic side chain. The modification was placed on the position of 1, 3, 4, 5, 8 and 9, e.g. D- Trp¹, Tyr¹, Tyr¹ Phe⁵, D-Trp¹ Phe⁸, Trp^3,4, D-Trp^1,3,4 Phe⁸, D-Trp3^,4 Phe⁸ D-Glu⁹, D-pF-Phe^3,4 Phe⁸ D-Glu⁹, Phe⁵, Glu⁵ Asp⁹, Tyr⁶ Asp⁹, Asp⁹ and Ala⁷-DSIP as well as Trp-Ala-Gly-Gly-Asp, Trp-Ala-Gly-Gly-Glu and Trp-Gly-Glu. DSIP and Phe⁵-DSIP were also prepared by pentafluorophenyl ester method. The purity of the synthetic peptides was checked by amino acid analysis, elemental analysis, thin layer chromatography and paper electrophoresis. The biological assay showed that the analogs of D-Trp¹, Tyr¹, Tyr¹ Phe⁵ and Ala⁷-DSIP as well as three short peptides were inactive whereas Phe⁵-DSIP showed similar activity as that of synthetic DSIP.