Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

Interaction in the molten system Rb2O‐P2O5‐TiO2‐NiO. Crystal structure of the langbeinite‐related Rb2Ni0.5Ti1.5(PO4)3

The interaction in the molten system Rb₂O‐P₂O₅‐TiO₂‐NiO was investigated at different molar ratios Rb/P = 0.5‐1.3, fixed Ti/P = 0.15, Ti/Ni = 1.0 at temperature range 1073–953 K. The conditions of formation of complex phosphates RbTi₂(PO₄)₃, Rb₂Ni_0.5Ti_1.5(PO₄)₃ and RbNiPO₄ have been determined. The new phosphate Rb₂Ni_0.5Ti_1.5(PO₄)₃ (space group P2₁3, a = 9.9386(2) Å) has been obtained and investigated by the single crystal X‐ray diffraction and FTIR‐spectroscopy. It has langbeinite‐like structure, that is built up from mixed (Ni/Ti)O₆‐octahedra and РО₄‐tetrahedra. Rubidium atoms are located in closed cavities of 3D‐framework.     

Pyranoside‐into‐Furanoside Rearrangement: New Reaction in Carbohydrate Chemistry and Its Application in Oligosaccharide Synthesis

Great interest in natural furanoside‐containing compounds has challenged the development of preparative methods for their synthesis. Herein a novel reaction in carbohydrate chemistry, namely a pyranoside‐into‐furanoside (PIF) rearrangement permitting the transformation of selectively O‐substituted pyranosides into the corresponding furanosides is reported. The discovered process includes acid‐promoted sulfation accompanied by rearrangement of the pyranoside ring into a furanoside ring followed by solvolytic O‐desulfation. This process, which has no analogy in organic chemistry, was shown to be a very useful tool for the synthesis of furanoside‐containing complex oligosaccharides, which was demonstrated by synthesizing disaccharide derivatives α‐D ‐Galp‐(1→3)‐β‐D ‐ Galf ‐OPr, 3‐O‐s ‐lactyl‐β‐D ‐ Galf ‐(1→3)‐β‐D ‐Glcp‐OPr, and α‐L ‐ Fucf ‐(1→4)‐β‐D ‐GlcpA‐OPr related to polysaccharides from the bacteria Klebsiella pneumoniae and Enterococcus faecalis and the brown seaweed Chordaria flagelliformis.     

Controlled Generation of 9-Boratriptycene by Lewis Adduct Dissociation: Accessing a Non-Planar Triarylborane

A highly bent triarylborane, 9-boratriptycene, was generated in solution by selective protodeboronation of the corresponding tetra-aryl boron ate complex with the strong Brønsted acid HNTf₂. The iptycene core confers enhanced Lewis acidity to 9-boratriptycene, making it unique in terms of structure and reactivity. We studied the stereoelectronic properties of 9-boratriptycene by quantifying its association with small N- and O-centered Lewis bases, as well as with sterically hindered phosphines. The resultant Lewis adducts exhibited unique structural, spectroscopic, and photophysical properties. Beyond the high pyramidalization of the 9-boratriptycene scaffold and its low reorganization energy upon Lewis base coordination, quantum chemical calculations revealed that the absence of π donation from the triptycene aryl rings to the boron vacant p_z orbital is one of the main reasons for its high Lewis acidity.     

σ-Aromaticity-Induced Stabilization of Heterometallic Supertetrahedral Clusters [Zn6Ge16]4− and [Cd6Ge16]4−

In this work, the largest heterometallic supertetrahedral clusters, [Zn₆Ge₁₆]⁴⁻ and [Cd₆Ge₁₆]⁴⁻, were directly self-assembled through highly-charged [Ge₄]⁴⁻ units and transition metal cations, in which 3-center–2-electron σ bonding in Ge₂Zn or Ge₂Cd triangles plays a vital role in the stabilization of the whole structure. The cluster structures have an open framework with a large central cavity of diameter 4.6 Å for Zn and 5.0 Å for Cd, respectively. Time-dependent HRESI-MS spectra show that the larger clusters grow from smaller components with a single [Ge₄]⁴⁻ and ZnMes₂ units. Calculations performed at the DFT level indicate a very large HOMO–LUMO energy gap in [M₆Ge₁₆]⁴⁻ (2.22 eV), suggesting high kinetic stability that may offer opportunities in materials science. These observations offer a new strategy for the assembly of heterometallic clusters with high symmetry.     

Stability,electronic, and optical properties of two-dimensional phosphoborane

The structure and properties of two-dimensional phosphoborane sheets were computationally investigated using Density Functional Theory calculations. The calculated phonon spectrum and band structure point to dynamic stability and allowed characterization of the predicted two-dimensional material as a direct-gap semiconductor with a band gap of ~1.5 eV. The calculation of the optical properties showed that the two-dimensional material has a relatively small absorptivity coefficient. The parameters of the mechanical properties characterize the two-dimensional phosphoborane as a relatively soft material, similar to the monolayer of MoS₂. Assessment of thermal stability by the method of molecular dynamics indicates sufficient stability of the predicted material, which makes it possible to observe it experimentally.     

The ESIS-RFQ channel versions: Advantages and disadvantages

The results of a numerical simulation of Au ion-beam transportation from an ESIS source to the beginning of the linear accelerator (radio frequency quadrupole (RFQ)) are given. Features of two versions of the channel (without separation and with a separation of charges) are analyzed.