Zeta-nonlocal scalar fields

We consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued Riemann zeta function through the d’Alembertian □ in its argument. The construction of the corresponding Lagrangians L starts with the exact Lagrangian for the effective field of the p-adic tachyon string, which is generalized by replacing p with an arbitrary natural number n and then summing over all n. We obtain several basic classical properties of these fields. In particular, we study some solutions of the equations of motion and their tachyon spectra. The field theory with Riemann zeta-function dynamics is also interesting in itself. Dedicated to Vasilii Sergeevich Vladimirov on his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 364–372, December, 2008.     

The evaluation of character Euler double sums

Euler considered sums of the form

Calculations of Isoelectronic Series of He Using Noninteger n‐Slater Type Orbitals in Single and Double Zeta Approximations

Using integer and noninteger n-Slater type orbitals in single- and double-zeta approximations, the Hartree-Fock-Roothaan calculations were performed for the ground states of first ten cationic members of the isoelectronic series of He atom. All the noninteger parameters and orbital exponents were fully optimized. In the case of noninteger n-Slater type orbitals in double zeta basis sets, the results of calculations obtained are more close to the numerical Hatree-Fock values and the average deviations of our ground state energies do not exceed 2×10^－6 hartrees of their numerical results.     

Size Fractionation of Two‐Dimensional Sub‐Nanometer Thin Manganese Dioxide Crystals towards Superior Urea Electrocatalytic Conversion

A universal technique has been proposed to sort two‐dimensional (2D) sub‐nanometer thin crystals (manganese dioxide MnO₂ and molybdenum disulfide MoS₂) according to their lateral dimensions. This technique is based on tuning the zeta potential of their aqueous dispersions which induces the selective sedimentation of large‐sized 2D crystals and leaves the small‐sized counterparts in suspension. The electrocatalytic properties of as‐obtained 2D ultrathin crystals are strongly dependent on their lateral size. As a proof‐of‐concept study, the small‐sized MnO₂ nanocrystals were tested as the electrocatalysts for the urea‐oxidation reaction (UOR), which showed outstanding performance in both half reaction and full electrolytic cell. A mechanism study reveals the enhanced performance is associated with the remarkable structural properties of MnO₂ including ultrathin (ca. 0.95 nm), laterally small‐sized (50–200 nm), and highly exposed active centers.     

The zeta potential of kaolin suspensions measured by electrophoresis and electroacoustics

The zeta potentials of kaolin dilute and concentrated suspensions were monitored using the techniques of electrophoresis and electroacoustics, respectively. The effect of addition of salt (KCl), a polymer material (Triton X-100), and an anionic surfactant (sodium dodecyl sulphate, SDS) on the suspension properties was investigated by electrophoresis. Electroacoustics was employed for the measurement of zeta potentials for the highest possible kaolin content in suspension and the effect of dilution. The effect of aging of a freshly prepared sample and kaolin isoelectric point was also studied. Using both techniques it was noted that there was no isoelectric point, just a maximum value in the magnitude of the kaolin suspension zeta potential. These maxima were observed also in the presence of Triton X-100 and SDS. An increase of the concentration of KCl and SDS in suspension shifted the maxima towards more acidic values, while in the presence of Triton X-100 the position of the zeta potential maxima remained constant. Electroacoustic techniques revealed that a freshly prepared concentrated suspension requires about six hours to equilibrate to achieve a steady zeta potential. Diluting the concentrated suspensions led to decrease of the zeta potential as ions bound to the surface desorbed and screened the surface charge. The zeta potential maxima remained unchanged even after heating the powder in an oven at 200°C (to remove any organic material) thereby suggesting that the most likely explanation for the maxima is isomorphic substitution.     

A cyclic group action on resolutions of quadruple systems

We prove that the necessary conditions for a quadruple system with a cyclic resolution to exist are sufficient for all possible v and all .     

On irregular prime power divisors of the Bernoulli numbers

Let ( ) denote the usual th Bernoulli number. Let be a positive even integer where or . It is well known that the numerator of the reduced quotient is a product of powers of irregular primes. Let be an irregular pair with . We show that for every the congruence has a unique solution where and . The sequence defines a -adic integer which is a zero of a certain -adic zeta function originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) -adic expansion of for irregular pairs with below 1000.

     

On Some Asymptotic Formulæ of Ramanujan

Let σ_b(n) ≔ Σ_d|nd^b. For every integer r ≥ 2 and for 0 < a < 1 we estabish an asymptotic estimate of Σ_{n ≤ x}σ^r_–a(n).     

On the quasi-derivation relation for multiple zeta values

Recently, Masanobu Kaneko introduced a conjecture on an extension of the derivation relation for multiple zeta values. The goal of the present paper is to present a proof of the conjecture by reducing it to a class of relations for multiple zeta values studied by Kawashima. In addition, some algebraic aspects of the quasi-derivation operator on Q〈x,y〉, which was defined by modeling a Hopf algebra developed by Connes and Moscovici, will be presented.