Notes on von Neumann–Jordan and James Constants for Absolute Norms on {\mathbb{R}^2}

Let ${\|\cdot\|_{\psi}}$ be the absolute norm on ${\mathbb{R}^2}$ corresponding to a convex function ${\psi}$ on [0, 1] and ${C_{\text{NJ}}(\|\cdot\|_{\psi})}$ its von Neumann–Jordan constant. It is known that ${\max \{M_1^2, M_2^2\} \leq C_{\text{NJ}}(\| \cdot \|_{\psi}) \leq M_1^2 M_2^2}$ , where ${M_1 = \max_{0 \leq t \leq 1} \psi(t)/ \psi_2(t)}$ , ${M_2 = \max_{0\leq t \leq 1} \psi_2(t)/ \psi(t)}$ and ${\psi_2}$ is the corresponding function to the ? ₂-norm. In this paper, we shall present a necessary and sufficient condition for the above right side inequality to attain equality. A corollary, which is valid for the complex case, will cover a couple of previous results. Similar results for the James constant will be presented.     

Unified Approach to Stabilization of Waves on Compact Surfaces by Simultaneous Interior and Boundary Feedbacks of Unrestricted Growth

Let $\mathcal{M}\subset\mathbb{R}^{3}$ be an oriented compact surface on which we consider the system: $$\left \{ \begin{array}{l@{\quad}l} u_{tt} - \Delta_{\mathcal{M}} u + a(x)g_{0}(u_{t})=0 & \text{in } \mathcal{M}\times\mathopen{]} 0,\infty[ ,\\ \partial_{\nu_{co}}u +u + b(x)g(u_t)=0 & \text{on } \partial \mathcal {M}\times\mathopen{]}0,+\infty[. \end{array} \right . $$ If $\mathcal{M}$ along with the localizers a, b and the nonlinear feedbacks g,g ₀ satisfy certain conditions then uniform (but not necessarily exponential) decay rates of the finite energy of solutions can be established. We present a unified approach that bridges and extends a number of earlier results on stabilization of 2nd-order hyperbolic equations on manifolds. The methodology captures geometric requirements for damping acting simultaneously on subsets of the interior and of the boundary, and shows how placements of these feedbacks can complement each other depending on the underlying surface. In addition, the results conveniently incorporate the existing theory that allows elimination of geometric conditions from the controlled boundary (in absence of nearby interior damping), and elimination of damping entirely from certain boundary neighborhoods. The model also admits feedbacks that grow sub- or super-linearly not only at the origin, but also at infinity and demonstrates an interplay between the regularity of solutions and asymptotic energy decay rates.     

Three new nortriterpene glycosides and two new triterpene glycosides from the bulbs of Scilla scilloides

Three new norlanostane-type triterpene glycosides, scillanostasides A, B, and C, and two new lanostane-type triterpene glycosides, scillanostasides D and E, were isolated from the bulbs of Scilla scilloides Druce (Liliaceae) along with one known norlanostane-type triterpene heptaglycoside, scillascilloside G-1. Their chemical structures were determined on the basis of spectroscopic data as well as chemical evidence.     

10 nm Spatial Resolution Fluorescence Imaging of Single Molecules by Near-Field Scanning Optical Microscopy Using a Tiny Aperture Probe

We demonstrate high-resolution fluorescence imaging of single molecules using near-field scanning optical microscopy (NSOM) with a tiny aperture probe for two different wavelengths in visible range in the illumination mode of operation. The spatial resolutions obtained at both excitation wavelengths were almost the same and the highest resolution realized was about 10 nm. To discuss the achievable resolution in aperture NSOM, we also employed a computer simulation by the finite-difference time-domain method for various aperture sizes and wavelengths. The resolution of 10 nm is predicted to be contributed by the single peak of localized near-field light around the rim of the aperture.     

High-pressure synthesis and structures of lanthanide germanides of LnGe5 (Ln=Ce, Pr, Nd, and Sm) isotypic with LaGe5

A series of lanthanide penta-germanides LnGe₅ (Ln=Ce, Pr, Nd and Sm) has been prepared by high-pressure (5–13 GPa) and high-temperature (500–1200 °C) reaction. CeGe₅ crystallizes in an orthorhombic unit cell (S.G. Immm (71)) with a=4.000(5) Å, b=6.192(5) Å, c=9.86(1) Å, and V=244.1(5) Å³. The new germanides are isotypic with LaGe₅ consisting of a Ge covalent network with tunnels where guest ions Ln³⁺ are situated. The network is composed of sublayers with edge-sharing Ge six-membered rings with only boat conformation. The sublayers are connected by rare eight-coordinated Ge atoms. The cell volume of the compounds systematically decreases from La to Sm compounds, except for CeGe_5, owing to the lanthanide contraction. The lattice constants of CeGe₅ are smaller than those of the Pr compound because it contains Ce⁴⁺ ions. CeGe₅ is paramagnetic above 2 K, but does not obey the Curie–Weiss law. PrGe₅ and NdGe₅ are Curie–Weiss type paramagnets with Weiss temperatures of –3.3 and –18.4 K. SmGe₅ shows an antiferromagnetic transition at 10.4 K.     

Formation of frozen α-AgI in superionic glass matrices at ambient temperature by rapid quenching

Superionic conducting phase of α-AgI, which is thermodynamically stable only above 147°C, was successfully frozen at ambient temperature in various kinds of glass matrices in the systems AgI---Ag₂O---M_xO_y (M_xO_y=B₂O₃, GeO₂,P₂O₅ and MoO₃) by a rapid quenching technique. The larger composition regions where α-AgI was frozen were observed in the systems with B₂O₃ and GeO₂ than in the systems with P₂O₅ and MoO₃. Since the glasses with B₂O₃ and GeO₂ have higher glass transition temperature than those with P₂O₅ and MoO₃, the former glasses have higher viscosities at 147° C. The former matrix glasses have stronger effect to depress the α-β transformation of AgI rather than the latter glasses in the course of rapid quenching of melts.     

On the robust estimation in poisson processes with periodic intensities

Under some regularity conditions, it is well known that the maximum likelihood estimator (MLE) is asymptotically normal and efficient. However, if the observation is contaminated, the MLE is not always an appropriate estimator. In this paper, we treat M-estimators and study their asymptotic behavior. By choosing estimation equations, robust M-estimators are presented for phase parameters.