Smarandache kn数字数列及其一类均值性质

Vk∈ N+,l＜k≤9,数列{a(k,n)}称作Smarandache kn数字数列,如果该数列中的每一个数都可以分成两部分,那么第二部分是第一部分的k倍,例如3n数字数列{α(3,n)}定义为{13,26,39,412,515,618,721,824,…}.利用初等及组合方法研究Smarandache kn数字数列...     

关于伪Smarandache函数的一个方程

任意n∈N＋,伪Smarandache无平方因子函数Zw（n）定义为最小的正整数m,满足n｜m^n．Z（n）定义为最小的正整数k,满足n｜（k（k＋1））/2．用初等方法研究了方程Zw（Z（n））-Z（Zw（n））=0的可解性,并证明了该方程有无穷多个正整数解．同时给出了不等式Zw（Z（n））-Z（Zw（n））〈0和Zw（Z（n））-Z（Zw（n））〉0的正整数解．     

关于指数Diophantine方程x^2＋2^（2m）=y^n

运用Lucas数本原素因数存在性的结果讨论方程x^2＋2^2m=y^n的正整数解（x,y,m,n）,证明了该方程仅有正整数解（x,y,m,n）=（2^（n-1）/2,2^r（rs-1）/2,s）和（2^3k.11,2^2k.5,3k＋1,3）适合n〉2,其中r和s是适合s〉2的正奇数,k是非负整数.     

一个关于SmarandacheLCM对偶函数的方程

橙 n ∈ N＋,著名的Smarandache LCM 函数的对偶函数定义为 SL ＊（n）＝ max｛k｜[1,2,？,k]｜ n ,k∈ N＋｝,Ω（n）表示n的所有素因子的个数。利用初等数论和分类讨论的方法研究了一个包含SL ＊（n）及素因子函数方程∑d｜n 1SL＊（d）＝Ω（n）的可解性,并给出了这个方程的所有正整数解的具体形式。     

数论函数方程φ（x）=S（x~k）的非平凡解

对于正整数n,设φ（n）和S（n）分别是Euler函数和Smarandache函数.对于给定的正整数k,如果正整数x适合x〉1以及φ（x）=S（xk）,则称x是方程φ（x）=S（xk）的非平凡解.运用初等数论方法证明了：（ⅰ）方程φ（x）=S（xk）的平凡解x都满足2k     

一般指数除数函数的渐近公式

令τ（e）（n）表示 n的指数除数个数。类似于 d（n）扩展到 dk （n）,τ（e）（n）也可以扩展到τk （e）（ n）。本文给出了τ3（ e）（ n）在小区间上的渐近公式。     

关于e-因子函数的一个渐近公式

（A）n∈N＋,设n=n=p1^a1P2^a1…pk^ak为n的标准素因数分解式,如果对于m=p1^β1P2^β2 …pr^βr有βi｜αi （I=1,2,...,k）,则称m为n的e-因子.令de（n）表示n的所有e-因子的个数.研究了k-full数集合上函数de（n）的均值性质,并得到了一个有趣的渐近公式.     

关于平面中一类特定数字集谱测度的探讨

讨论平面上由整数扩张矩阵M=[a b d c],det（M）=ac-bd∈2Z和数字集D={（00）,（m0）,（nk1）,（lk2）}（m≠0,n,l∈Ζ,k1,k2∈2Ζ＋1）决定的L2（μM,D）中无限正交系的存在性,及由M=（2 b 0 2）,b∈Z,D={（00）,（10）,（01）,（k1k2）}决定的μM,D是否是谱测度. 更多还原     

关于SmarandacheCeil函数的一类均值问题

研究了SmarandacheCeil函数与素因子积函数U（咒）的均方值的分布问题．利用解析方法给出了（Sa（n）-U（n））2。的一个有趣的渐近公式，其中k≥2，以为自然数．     

关于无k次幂因子数的上界估计

设k≥2为给定的正整数,n>1为任意自然数.如果m∈N ,m>1有mk n,则称n为无k次幂因子数,特别当k=2,3时,称n为无平方因子数及无立方因子数.利用初等方法研究无k次幂因子数的性质,并获得第n个无k次幂因子数的一个较强的上界估计.     

广义矩阵迹的贝尔曼不等式

引入了广义矩阵迹τ：Mn(C(Ω））→C（Ω），讨论了在广义矩阵迹下的贝尔曼不等式。证明了C^*-代数Mn(C(Ω))中任意两个正元A，B及A k∈N，有τ（（AB）^k)≤τ（A^kB^k),这便在更一般的框架下给出了Bellman问题的一个肯定问题。同时还利用C^*-代数证明了其它一些相关不等式。     

Reduction of hexavalent chromium by H2O2 in acidic solutions

The rates of the reduction of Cr(VI) with H2O2 were measured in NaCl solutions as a function of pH (1.5-4.8), temperature (5-40 degrees C), and ionic strength (I = 0.01-2 M) in the presence of an excess of reductant. The rate of Cr(VI) reduction is described by the general expression -d[Cr(VI)]/dt = k2[Cr(VI)](m)[H2O2](n)[H+](z), where m = 1 and n and z are two interdependent variables. The value of n is a function of pH between 2 and 4 (n = (3 x 10(a))/(1 + 10(a)), where a = -0.25 - 0.58pH + 0.26pH2) leveling off at pH < 2 (where n approximately = 1) and pH > 4 (where n approximately = 3). The rates of Cr(VI) reduction are acid-catalyzed, and the kinetic order z varies from about 1.8-0.5 with increasing H2O2 concentration, according to the equation z = 1.85 - 350.1H2O2 (M) which is valid for [H2O2] < 0.004 M. The values of k2 (M(-(n+z)) min(-1)) are given by k2 = k/[H+](z) = k1/[H2O2](n)[H+](z), where k is the overall rate constant (M(-n) min(-1)) and k, is the pseudo-first-order rate constant (min(-1)). The values of k in the pH range 2-4 have been fitted to the equation log k = 2.14pH - 2.81 with sigma = +/- 0.18. The values of k2 are dependent on pH as well. Most of the results with H2O2 < 3 mM are described by log k2 = 2.87pH - 0.55 with sigma = +/- 0.54. Experimental results suggest that the reduction of Cr(VI) to Cr(III) is controlled by the formation of Cr(V) intermediates. Values of k2 and k calculated from the above equations can be used to evaluate the rates of the reaction in acidic solutions under a wide range of experimental conditions, because the rates are independent of ionic strength, temperature, major ions, and micromolar levels of trace metals (Cu2+, Ni2+, Pb2+). The application of this rate law to environmental conditions suggests that this reaction may have a role in acidic solutions (aerosols and fog droplets) in the presence of high micromolar concentrations of H2O2.     

Water-absorption rate equation of rice for brewing sake

This study was undertaken to analyze the kinetics of water absorption and to derive an equation for the rate at which water is absorbed by rice for brewing sake. We used two rice varieties: Gin-oumi, commonly used as a staple food, and Gohyakumangoku, a variety used particularly for brewing sake. The water-absorption rate equations of Gin-oumi and Gohyakumangoku were postulated based on the following equations. For Gin-oumi (water content, 11.5%), dx/dtheta=k(1-x)(n), n=1, k=(2 x 10(-9))exp(0.0604 x (t+273.15)). For Gohyakumangoku (11.5%), dx/dtheta=k(1-x)(n)(x+a), n=1, a=0.29, k=(2 x 10(-8))exp(0.0534 x (t+273.15)). Here, x, theta (min), and t ( degrees C) are the water absorbing ratio, time, and temperature, respectively. The result shows that the values of the temperature-dependence parameter k (min(-1)), as well as its curves, are different; a typical rice grain has a monotonically smooth curve, whereas that suitable for sake brewing has an S-shaped curve.     

On the nonlinear relationship between k(obs) and reductant mass loading in iron batch systems

A first-order relationship between reaction rate and reductant loading is commonly invoked (but rarely verified) for granular iron batch systems. In this study, a linear relationship between the pseudo-first-order rate constant (k(obs)) for polyhalogenated alkane reduction and iron mass loading (rhom) was only obtained when reduction appeared to be mass-transport-limited. For most alkyl polyhalides, a nonlinear relationship was observed under various experimental conditions, causing surface-area-normalized rate constants (k(SA)) to decrease by as much as an order of magnitude with increasing rhom. While a detailed explanation for this nonlinearity remains unclear, the reaction order in rhom (from plots of log(k(obs)) versus log(rhom)) approached unity as the system pH decreased, suggesting that the behavior may be linked to changes in the thickness of a passive oxide overlayer. We obtained strong linear correlations between k(obs) values and the concentration of aqueous iron(ll) generated over a 24 h period in batch systems with that same rhom. Equally strong correlations were obtained when k(obs) was related to the summed concentration of protons and water reduced over an equivalent time interval at that same rhom. Such correlations suggest a possible link between those sites responsible for proton/water reduction and the surface species participating in alkyl polyhalide reduction by granular iron.     

Biotic and abiotic reduction and solubilization of Pu(IV)O₂•xH₂O(am) as affected by anthraquinone-2,6-disulfonate (AQDS) and ethylenediaminetetraacetate (EDTA)

This study measured reductive solubilization of plutonium(IV) hydrous oxide (Pu(IV)O(2)·xH(2)O((am))) with hydrogen (H(2)) as electron donor, in the presence or absence of dissimilatory metal-reducing bacteria (DMRB), anthraquinone-2,6-disulfonate (AQDS), and ethylenediaminetetraacetate (EDTA). In PIPES buffer at pH 7 with excess H(2), Shewanella oneidensis and Geobacter sulfurreducens both solubilized <0.001% of 0.5 mM Pu(IV)O(2)·xH(2)O((am)) over 8 days, with or without AQDS. However, Pu((aq)) increased by an order of magnitude in some treatments, and increases in solubility were associated with production of Pu(III)((aq)). The solid phase of these treatments contained Pu(III)(OH)(3(am)), with more in the DMRB treatments compared with abiotic controls. In the presence of EDTA and AQDS, PuO(2)·xH(2)O((am)) was completely solubilized by S. oneidensis and G. sulfurreducens in ～24 h. Without AQDS, bioreductive solubilization was slower (～22 days) and less extensive (～83-94%). In the absence of DMRB, EDTA facilitated reductive solubilization of 89% (without AQDS) to 98% (with AQDS) of the added PuO(2)·xH(2)O((am)) over 418 days. An in vitro assay demonstrated electron transfer to PuO(2)·xH(2)O((am)) from the S. oneidensis outer-membrane c-type cytochrome MtrC. Our results (1) suggest that PuO(2)·xH(2)O((am)) reductive solubilization may be important in reducing environments, especially in the presence of complexing ligands and electron shuttles, (2) highlight the environmental importance of polynuclear, colloidal Pu, (3) provide additional evidence that Pu(III)-EDTA is a more likely mobile form of Pu than Pu(IV)-EDTA, and (4) provide another example of outer-membrane cytochromes and electron-shuttling compounds facilitating bioreduction of insoluble electron acceptors in geologic environments.     

In vitro skin permeation of sunscreen agents from O/W emulsions

The effects of different emulsifiers on the in vitro permeation through human skin of two sunscreen agents [octylmethoxycinnamate (OMC) and butylmethoxydibenzoylmethane (BMBM)] were investigated from O/W emulsions. The test formulations were prepared using the same oil and aqueous phase ingredients and the following emulsifier and coemulsifier systems: Emulgade SE((R)) (ceteareth-12 and ceteareth-20 and cetearyl alcohol and cetyl palmitate) and glycerylmonostearate (emulsion 1); Brij 72((R)) (steareth-2), Brij 721((R)) (steareth-21) and cetearyl alcohol (emulsion 2); Phytocream((R)) (potassium palmitoyl-hydrolysed wheat protein and glyceryl stearate and cetearyl alcohol) and glycerylmonostearate (emulsion 3); Montanov 68((R)) (cetearyl glucoside and cetearyl alcohol) (emulsion 4); Xalifin-15((R)) (C(15-20) acid PEG-8 ester) and cetearyl alcohol (emulsion 5). The cumulative amount of OMC that permeated in vitro through human skin after 22 h from the formulations being tested decreased in the order 3 > 1 congruent with 4 > 5 > 2 and was about nine-fold higher from emulsion 3 compared with that from emulsion 2. As regards BMBM, no significant difference was observed as regards its skin permeation from emulsions 1, 3, 4 and 5, whereas formulation 2 allowed significantly lower amounts of BMBM to permeate the skin. In vitro release experiments of OMC and BMBM from emulsions 1-6 through cellulose acetate membranes showed that only emulsions 4 and 5 provided pseudo-first-order release rates only for OMC. The results of this study suggest that the type of emulsifying systems used to prepare an O/W emulsion may strongly affect sunscreen skin permeation from these formulations. Therefore, the vehicle effects should be carefully considered in the formulation of sunscreen products.     

包含Smarandache幂函数的均值问题

利用初等数论和解析数论的方法研究著名的Smarandache幂函数SP(n)的均值估计问题.首先给出Smarandache幂函数SP(n)的定义,几个重要的性质和相关引理.在此基础上得到了一些有意义的结果,即在简单数序列上得到了∑n∈A n≤x1/S(SP(n))和∑n∈A n≤x S(SP(n))的均值.     

Formate ion decomposition in water under UV irradiation at 253.7 nm

Formate ion (HCO2-) occurs in natural waters as a result of photooxidation of humic substances. Under UV irradiation, as applied in water purification (253.7 nm), formate ion decomposed following split-rate pseudo-zero-order kinetics (k1 and k2 are initial and final rate constants, respectively). In the presence of dissolved oxygen (DO), it was found that (a) k1 < k2, (b) k1 and k2 increased with initial formate ion concentration ([HCO2-]0 = (1.73-38.3) x 10(-5) mol L(-1)) and absorbed UV intensity (Ia = (1.38-3.99) x 10(-6) mol quanta L(-1) s(-1)), and (c) k1 and k2 were relatively insensitive to initial pH (pHo = 5.41-8.97) in buffer-free solutions. Both rate constants decreased with increasing carbonate alkalinity ((0-1.0) x 10(-3) mol L(-1)) and k1 was virtually unchanged in phosphate buffer at pH0 between 5.25 and 9.92. Carbonate buffer lowered the rate of formate ion decay, possibly due to scavenging of OH* radicals. Initial rate constant k1 slightly increased with temperature (15-35 degrees C), while k2 remained unchanged. The reaction pH increased rapidly during irradiation of buffer-free NaHCO2 solution to approach an equilibrium level as [HCO2-] reached the method detection level (MDL). The pH profile of buffer-free formate ion decay was estimated using closed-system equilibrium analysis. DO utilization during UV irradiation was 0.5 mol of O2/mol of HCO2-, while nonpurgeable organic carbon (NPOC) measurements on kinetic samples closely followed the HCO2- profile, thus strongly suggesting the transformation of HCO2- -C to CO2 in the presence of DO. In DO-free water, k1 > k2 was observed. Furthermore, k(1,DO FREE) > k(1,DO) (k(1,DO) = k1) and k(2,DO FREE) < k(2,DO) (k(2,DO) = k2). The effect of dual acid solutions on HCO2- decay was examined in a mixture of NaHCO2 and sodium oxalate (Na2C2O4). HCO2- decomposed readily until [HCO2-] approximately equal to MDL but at a lower rate than in buffer-free HCO2- solutions, while C2O4(2-) remained virtually unchanged. C2O4(2-) decay commenced following near complete conversion of HCO2-.     

关于双曲函数方幂之和

设shx,chx和sinβ,cosβ是双曲正、余弦函数和三角函数,用发生函数的方法得到双曲正,余弦函数方幂与等比序列乘积之和n∑k=0 dkshrkx,n∑k=0 dk chrkx和双曲正、余弦函数带有三角函数方幂n∑k=0 shrkxsinkβ,n∑k=0 shrkxcoskβ的计算公式.