Transcendence of the values of infinite products in several variables

The aim of this paper is to prove the transcendence of certain infinite products. As applications, we get necessary and sufficient conditions for transcendence of the value of $\Pi_{k=0}^{\infty}(1+a_{k}^{(1)}{z_{1}r^{k}}+\cdot\cdot\cdot+a_{k}^{(m)}{z_{m}r^{k}})$ at appropriate algebraic points, where r ≥ 2 is an integer and {a_n ⁽ⁱ⁾}_{n≥ 0} (1 ≤ i ≤ m) are suitable sequences of algebraic numbers.     

Infinitely many solutions for a nonlinear Schrödinger equation with general nonlinearity

We prove the existence of infinitely many solutions for

$$\begin{aligned} - \Delta u + V(x) u = f(u) \quad \text { in } \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \end{aligned}$$

where V(x) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions on f(u) only around 0 and at \(\infty \).

     

Study of Strangeness Photoproduction on the Neutron in the Threshold Region

Photoproduction of the neutral kaon on the deuteron has been investigated at the Research Center for Electron Photon Science, Tohoku University. We constructed the Neutral Kaon Spectrometer-2 for the detection of charged particles from the decay of the neutral kaon and the hyperon. We obtained a momentum distribution of K ⁰ with the inclusive measurement. It was consistent with the previous measurement. The total cross section of γ + d → K ⁰ + Λ + p was estimated from the measured integral cross section of γ + d → Λ + X. The total cross section with respect to the photon energy was compared with the theoretical calculations. It favored the Saclay-Lyon A model calculation with the ratio of the neutral to charged coupling constants of the axial-vector meson, K ₁, as ~ ?1.5. The energy dependence and the magnitude of the total cross section were similar to the total cross section of γ + p → K ⁺ Λ.     

On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system

The paper concerns multiplicity of vector solutions for nonlinear Schrödinger systems, in particular of semi-positive solutions. New variational techniques are developed to study the existence of this type of solutions. Asymptotic behaviors are examined in various parameter regimes including both attractive and repulsive cases.     

Basic [Au25(SCH2CH2Py)18]−⋅Na+ Clusters: Synthesis,Layered Crystallographic Arrangement,and Unique Surface Protonation

The synthesis of high‐purity and high‐yield Au₂₅ clusters protected by the basic pyridyl ethanethiol (HSCH₂CH₂Py, 4‐PyET and 2‐PyET) is presented. Single‐crystal X‐ray diffraction of the [Au₂₅(4‐PyET)₁₈]^??Na⁺ clusters has revealed a structure similar to that known for the phenyl ethanethiolate analogue, but with pyridyl‐N coordination to Na⁺, a more relaxed ligand shell, and a profoundly layered arrangement in the solid state. Because of the pendant Py moiety, the [Au₂₅(PyET)₁₈]^? clusters are endowed with unique (de)protonation equilibria, which has been characterized in detail by UV/Vis absorption and ¹H NMR spectroscopy. [Au₂₅(PyET)₁₈]^? clusters showed an unexpectedly H⁺‐dependent solubility that is tunable in aqueous and organic solvents. The successful synthesis of the basic Py‐terminated thiolate‐protected Au₂₅ clusters paves the way to realize a new family of metalloid clusters possessing basic properties.     

Peripheral Modifications of meso-Hydroxyporphyrins: Formation of π-Electronic Anions and Ion-Pairing Assemblies

Phenyl groups were introduced at the β-positions proximal to the meso-hydroxy moiety in hydroxyporphyrin Ni^II complexes by oxidized BINAP Pd^II complexes. Ion-pairing assemblies of deprotonated π-electronic anions, anionic site of which was stabilized by the introduced phenyl moieties with a bulky cation, were formed. They showed charge-by-charge assemblies, assembly modes of which were modulated by the anionic building units.