Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Gedankenexperiment for Modified ZPE and Planck’s “Constant”, h, in the Beginning of Cosmological Expansion, Partly Due to NLED

We initially look at a non singular universe representation of entropy, based in part on what was brought up by Muller and Lousto. This is a gateway to bringing up information and computational steps (as defined by Seth Lloyd) as to what would be available initially due to a modified ZPE formalism. The ZPE formalism is modified as due to Matt Visser’s alternation of k (maximum) ~ 1/(Planck length), with a specific initial density giving rise to initial information content which may permit fixing the initial Planck’s constant, h, which is pivotal to the setting of physical law. The settings of these parameters depend upon NLED.

Stoichiometry and Stability Constant Values for Copper (II) Chelates with Ethylene Diamine in Deep Eutectic Solvents (DES) (Ethaline) Solutions

In this study we used the deep eutectic solvents (ionic liquids) to investigate the reaction between copper (II) with ethylene diamine (en). Two of the existing methods for analyzing spectrophotometric measurements have been applied for establishing, the stoichiometry and whenever possible, the stability constants of the chelates formed. The method of continuous variations was necessary to determine first whether, the metal ion and the ligand ethylene diamine form one or more than one chelate, when more than one chelate formed, the results obtained depend on the wavelength and for meaningful conclusions the wavelengths were carefully selected. The empirical formulae of the chelates were further substantiated by the molar ratio method. The effect of time and temperature on the formation and stability of these chelates in solution is also studied. The stability constants, K1 and K2 for the copper (II) chelates were calculated, though reliable, and are comparable to literature values.

基于Euler/N-S方程的跨音速非线性静气动弹性问题研究

在C-H网格的基础上,采用Jam eson的中心差分有限体积法求解Eu ler/N-S方程,采用结构影响系数法计算结构的弹性变形,用三角元面积加权法和常体积转换法(CVT)实现流固耦合。

任意曲线坐标系Euler方程S2流面的计算方法

为提高S2流面计算方法对涡轮复杂几何的适应性和计算能力,详细推导了任意正交曲线坐标系下应用于流道中心S2流面计算的Euler方程,提出了任意曲线坐标系Euler方程S2流面的计算方法,发展了适用于三阶精度总变差减小的差分格式的数学模型,结合了隐格式时间推进、Riemann问题求解等技术。在应用于带弧形凸起的流道及某直叶栅气动参数计算,并对某三级低压涡轮进行性能预测的结果表明:所提方法对激波具有较高的捕捉精度,能够较准确地获得叶栅气动参数分布,且与实验结果吻合良好;对于多级涡轮总性能参数和气动参数分布均有较高的计算精度,是一种可为涡轮设计提供快速、可靠的计算方法。

Stieltjes常数的弱有界及其衍生Euler常数的数学表示

为了探讨Euler常数γ的数学表示式,通过对Stieltjes常数γk=nl→im∞S(Nk)=∑Nn=1lnknn-1k+1lnk+1(N+1)(k=0,1,2,…)的一个弱有界进行了进一步的优化估计,然后从该估计出发,把Euler常数γ的一个数学表达式γ=limx→0+{∑∞n=11n1+x-1x}的右边函数展成关于x的幂级数,并对其一致收敛性进行了详细地讨论.最后通过构造一个函数g(x)∑∞n=1(-1)n-1n1+x,(∞<1,x∈R)而得到Euler常数γ的一个新的数学表达式.

关于Euler函数的Makowski猜想

对于正整数n,设φ(n)是n的Euler函数.该文证明了:如果φ(n+3)=φ(n)+2,则n=2pr或2pr-3,其中p是适合p≡3(mod 4)的素数,r是正整数.

Some Kind of Equations Involving Euler＇s Function

For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.

Mass Constituents of a Flat Lattice Multiverse: Conclusion from Similarity between Two Universal Numbers, the Rocksalt-Type 2<i>D</i>Madelung Constant and the Golden Mean

In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean &phi;=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by &phi;, one reaches again 1 + &phi;as the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ&minus;1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number &phi;may indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant &pi;, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.

Henry's Law constant for phosphine in seawater:determination and assessment of influencing factors

The Henry＇s Law constant （k） for phosphine in seawater was determined by multiple phase equilibration combined with headspace gas chromatography. The effects of pH, temperature, and salinity on k were studied. The k value for phosphine in natural seawater was 6.415 at room temperature （approximately 23℃）. This value increases with increases in temperature and salinity, but no obvious change was observed at different pH levels. At the same temperature, there was no significant difference between the k for phosphine in natural seawater and that in artificial seawater. This implies that temperature and salinity are major determining factors for k in marine environment. Double linear regression with Henry＇s Law constants for phosphine as a function of temperature and salinity confirmed our observations. These results provide a basis for the measurement of trace phosphine concentrations in seawater, and will be helpful for future research on the status of phosphine in the oceanic biogeochemical cycle of phosphorus.

高阶熵条件格式下Euler方程与N-S方程的混合算法

针对在工程问题中使用N S方程计算量大的问题 ,提出只在粘性作用显著的局部区域求解N S方程 ,在流场的其余部分采用Euler方程计算的办法来模拟具有复杂外形的流动问题 ,并提出了一种基于高阶熵条件格式的算子分裂算法 .应用所构造的算法对绕迫击炮弹的亚、跨超声速流动进行了模拟计算 .计算结果与试验数据的对比表明 ,所提出的方法是切实可行的 .

Euler常数与Stirling序列的快速收敛法

对某些重要的收敛序列,借助几何直观方法估计了它们的收敛余项。

关于Landau常数和Euler-Mascheroni常数的渐近展开式以及Stirling级数的系数

文章给出一个递推关系式来确定Landau常数的渐近展开式的系数.考虑了Euler-Mascheroni常数和n!的渐近展开式,并给出了递推关系式来确定每个展开式的系数,没有利用Bernoulli数.

Reinvestigation of the Henry's law constant for hydrogen peroxide with temperature and acidity variation

Hydrogen peroxide is not only an important oxidant in itself; it also serves as both sink and temporary reservoir for other important oxidants including HOx （OH and HO2） radicals and O3 in the atmosphere. Its partitioning between gas and aqueous phases in the atmosphere, usually described by its Henry＇s law constant （KH）, significantly influences its role in atmospheric processes. Large discrepancies between the KH values reported in previous work, however, have created uncertainty for atmospheric modelers. Based on our newly developed online instrumentation, we have re-determined the temperature and acidity dependence of KH for hydrogen peroxide at an air pressure of （0.960 ± 0.013） atm （1 atm = 1.01325 × 10^5 Pa）. The results indicated that the temperature dependence of KH for hydrogen peroxide fits to the Van＇t Hoff equation form, expressed as lnKH = a/T - b, and a = -△H/R, where KH is in M/atm （M is mol/L）, T is in degrees Kelvin, R is the ideal gas constant, and AH is the standard heat of solution. For acidity dependence, results demonstrated that the KH value of hydrogen peroxide appeared to have no obvious dependence on decreasing pH level （from pH 7 to pH 1）. Combining the dependence of both temperature and acidity, the obtained a and b were 7024 ± 138 and 11.97 ± 0.48, respectively, AH was （58.40 ± 1.15） kJ/（K.mol）, and the uncertainties represent σ. Our determined KH values for hydrogen peroxide will therefore be of great use in atmospheric models.

Physical interpretation of Planck's constant based on the Maxwell theory

The discovery of the Planck relation is generally regarded as the starting point of quantum physics.Planck＇s constant h is now regarded as one of the most important universal constants.The physical nature of h,however,has not been well understood.It was originally suggested as a fitting constant to explain the black-body radiation.Although Planck had proposed a theoretical justification of h,he was never satisfied with that.To solve this outstanding problem,we use the Maxwell theory to directly calculate the energy and momentum of a radiation wave packet.We find that the energy of the wave packet is indeed proportional to its oscillation frequency.This allows us to derive the value of Planck＇s constant.Furthermore,we show that the emission and transmission of a photon follows the all-or-none principle.The ＂strength＂ of the wave packet can be characterized by ζ,which represents the integrated strength of the vector potential along a transverse axis.We reason that ζ should have a fixed cut-off value for all photons.Our results suggest that a wave packet can behave like a particle.This offers a simple explanation to the recent satellite observations that the cosmic microwave background follows closely the black-body radiation as predicted by Planck＇s law.

ASYMPTOTIC EQUIVALENCE OF ALTERNATELY ADVANCED AND DELAYED DIFFERENTIAL SYSTEMS WITH PIECEWISE CONSTANT GENERALIZED ARGUMENTS

In this paper, we investigate the existence, uniqueness and the asymptotic equiv- alence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type （DEPCAG）. This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.

Stability constants of Sb^5+with Cl−and thermodynamics of Sb−S−Cl−H2O system involving complex behavior of Sb with Cl

The stability constants of Sb^5+with Cl−as well as thermodynamics of the Sb−S−Cl−H2O system were calculated.The stability constants of Sb5+with Cl−were obtained by theoretical calculations of the absorbance of a Sb5+-containing solution at different Cl^−concentrations,which was detected by spectrophotometric analysis at certain wavelengths of light(380 nm).The logarithmic values versus 10 of stability constants of Sb^5+with Cl−were 1.795,3.150,4.191,4.955,5.427 and 5.511,respectively,and partly filled the data gaps in the hydrometallurgy of antimony.The presence and distribution of pentavalent antimony compounds under different conditions were analyzed based on equilibrium calculations.Thermodynamic equilibrium calculations were performed for Sb−S−Cl−H2O system,which included the complex behavior of Sb with Cl,and the equilibrium equations of related reactions in this system were integrated into the potential−pH diagram.

Some Remarks for the Relationships between the Generalized Bernoulli and Euler Polynomials

In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.

颅骨点云曲面重建的Euler’s Elastica模型

为从具有复杂拓扑关系的颅骨点云数据中重建精确、光滑的隐式曲面,提出了基于Euler’s Elastica的变分水平集方法.首先,将Euler’s Elastica作为正则化约束引入变分能量泛函,采用增广拉格朗日方法将变分能量泛函最小化问题转化为一系列子问题;其次,分别采用广义软阈值公式、解析公式和快速傅里叶变换求解;最后,在计算过程中采用投影算法,有效地避免了水平集函数的重新初始化过程.实验结果表明,相较于现有方法,所提基于Euler’s Elastica的变分水平集方法具有更好的收敛性和有效性,能够提高重建的精确性,避免曲面边缘的缺失.

关于Euler-Mascheroni常数的一个改进序列

文中对收敛到Euler-Mascheroni常数的定义序列进行了改进,使得序列收敛到Euler-Mascheroni常数的速度是给定的参数,并得到了关于该序列的一个双边不等式。