Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth

We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst–Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.     

Evaluation formulas for Tornheim's type of alternating double series

In this paper, we give some evaluation formulas for Tornheim's type of alternating series by an elementary and combinatorial calculation of the uniformly convergent series. Indeed, we list several formulas for them by means of Riemann's zeta values at positive integers.

     

An Electroacoustic Investigation of Concentrated Aqueous Suspensions of Calcium Pyrophosphate

The zeta potential of concentrated suspensions of calcium pyrophosphate were investigated using electroacoustics. The particles were negatively charged over the entire pH range studied. It proved impossible to reduce the pH of the suspension below 1.8, but the iso electric point could be estimated to occur at approximately pH 1. Two commercially available dispersants (one cationic and one anionic) were then added in small increments to the suspension in order to follow the change in zeta potential as the dispersants adsorbed onto the particles. From the shape of the curve it was possible to estimate the optimum adsorbed amount of dispersant required to fully coat the particles. The cationic dispersant adsorbed strongly and the optimum dosage was estimated at 2.5 ml of active polymer/kg of powder. Anionic dispersants are known to adsorb onto negatively charged ceramic oxide particles, but the anionic dispersant used in this study did not adsorb onto the negatively charged calcium pyrophosphate particles.     

Runge–Kutta interpolants for high precision computations

Runge–Kutta (RK) pairs furnish approximations of the solution of an initial value problem at discrete points in the interval of integration. Many techniques for enriching these methods with continuous approximations have been proposed. Here we construct C ¹ continuous, eighth and ninth order interpolation methods for a recently appeared RK pair of orders 9(8). These interpolants share a very small leading truncation error making them suitable for use at quadruple precision, i.e. 32–33 decimal digits of accuracy. Extended numerical results justify our effort.     

Riemann Zeta函数的六阶和

利用概率论与组合数学的方法,研究了与Riemann-zeta函数ξ(k)的部分和ξ_n(k)有关的一些级数,计算出了一些重要的和式．特别的,Euler的著名结果5ξ(4)= 2ξ~2(2)能够从四阶和式直接推出．因此,通过计算全部的11个六阶和式,研究它们之间的非平凡关系,就有可能得到ξ(3)的数值．     

Applications of inversion formulas to the joint t-universality of Lerch zeta functions

In this paper, we define λ-joint, a^′-joint, (λ,λ)-joint, (λ,a^′)-joint and (a^′,a^′)-joint t-universality of Lerch zeta functions and consider the relations among those. Next we show the existence of (λ,λ)-joint t-universality. Finally, we also show the existence of λ-joint, a^′-joint, (λ,a^′)-joint and (a^′,a^′)-joint t-universality by using inversion formulas.     

Asymptotic expansions of the Hurwitz-Lerch zeta function

The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables.     

Coupling between polymer adsorption and colloidal particle aggregation

Studies of the adsorption of high molecular weight polymers on colloidal latex and silica particles and their subsequent flocculation were carried out. Neutral polyethylene oxide samples with both a narrow and a broad molecular weight distribution were used together with low charged cationic copolymers. The influence of the particle concentration and polymer dose on the flocculation were systematically investigated under quiescent conditions.Equilibrium bridging only occurred with polyelectrolyte, even in very dilute suspensions, at high particle coverage. In contrast to this, non-equilibrium bridging occurred with both neutral polymer and polyelectrolytes but only for more concentrated suspensions and small amounts of adsorbed polymer. Polymer adsorption in dilute suspensions, which did not show particle aggregation was measured an electrophoretic technique. In more concentrated suspensions, where flocculation takes place, we found that aggregation prevents further polymer adsorption and induces both an excluded volume and a surface effect. The consequences on the shape of the isotherms differ according to the aggregation mechanism.A significant decrease of the amount, , of adsorbed polymer is observed with non-equilibrium bridging. When both mechanisms simultaneously contribute to the aggregation, the value of depends on their relative importance. In the intermediate range of copolymer dose their respective contributions are critically sensitive to the details of the mixing step and stirring, leading to non reproducible experimental results.