Sharp remainder of a critical Hardy inequality

An explicit representation is given to the remainder of a critical Hardy inequality in \({L^{n}(\mathbb{R}^{n})}\) with \({n \ge 2}\).     

Reproducibility of GABA measurements using 2D J-resolved magnetic resonance spectroscopy

We determined the reproducibility of GABA (gamma-aminobutyric acid) measurements using 2D J-resolved magnetic resonance spectroscopy (MRS) on a clinical 1.5-T MR imaging scanner. Two-dimensional J-resolved spectra were acquired in vitro across five GABA concentrations using a volume head coil and a 5-in. surface coil. Additional spectra using a sixth GABA phantom with a very low concentration and from a healthy volunteer were recorded in the 5-in. surface coil only. In each case, the 3.01-ppm GABA resonance was quantified; for comparison, the peak integrals of choline (3.2 ppm) and creatine (3.03 ppm) were recorded. At a physiological concentration (1.2 mM), in vitro GABA measurement was significantly more reproducible in the surface coil than in the volume coil (P=.005), with coefficients of variation (CVs) being less than 16% with the surface coil and up to 68% with the volume head coil. At the smallest concentration of in vivo GABA reported using other spectroscopy techniques (0.8 mM) and detected only using the surface coil, the CV for GABA was 23% and was less than 10% for choline and creatine, which compare favorably with results from published studies. In vivo, the CV for GABA measurement was 26%, suggesting that 2D J-resolved MRS would be suitable for detecting physiological changes in GABA, similar to those reported using other methods.     

Theoretical study on the molecular structures of X-, α-, and β-types of lithium phthalocyanine dimer

We report here the results from theoretical calculations of the potential energy curves, the geometry optimizations, and the electronic structures for three dimers of lithium phthalocyanine (LiPc) by using three types of functional systems: PBE1PBE, B3LYP, and M06. The results were discussed in comparison with those obtained for the dimers of magnesium phthalocyanine (MgPc). The long-range dispersive interactions were considered in part using these functional systems in the increasing order of PBE1PBE, B3LYP, and M06. The mechanism whereby the dispersive interactions affect the geometric and electronic structures of the LiPc and MgPc dimers is discussed. The calculated results provide insight into the computational methods for both open- and closed-shell metal phthalocyanine (MPc) dimers: Although the PBE1PBE and B3LYP functional systems cannot evaluate a weak dispersion interaction appropriately, the M06 functional can estimate a weak dispersion interaction well in both open- and closed-shell MPc dimers. Basis set superposition error (BSSE) corrections play an important role for the quantitative analysis; however, the calculation results without BSSE corrections may be sufficient for the qualitative discussion on the properties of these dimers such as geometries, stabilities, electronic structures, and so on.     

Metal array fabrication through self-assembly of Pt-complex-bound amino acids

A new type of Pt-complex-bound amino acid was synthesized by condensation of a cyclometalated Pt complex with the side-chain residue of N- and C-alkylated glutamic acid. Self-assembly of the Pt-bound lipophilic amino acid afforded a supramolecular gel in organic solvents, which comprised fibrous lamellar aggregates that supported a highly oriented Pt array.     

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ? ₊ ^N with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ?? 3, ?? > 0, 0 < T ?? ??, and u ₀ ?? C ₀(? ₊ ^N ). We prove the existence of a threshold number ?? _N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ? ₊ ^N .     

Twofold spontaneous symmetry breaking in the heavy-fermion superconductor UPt3

The field-orientation dependent thermal conductivity of the heavy-fermion superconductor UPt3 was measured down to very low temperatures and under magnetic fields throughout the distinct superconducting phases: B and C phases. In the C phase, a striking twofold oscillation of the thermal conductivity within the basal plane is resolved reflecting the superconducting gap structure with a line of node along the a axis. Moreover, we find an abrupt vanishing of the oscillation across a transition to the B phase, as a clear indication of a change of gap symmetries. We also identify extra two line nodes below and above the equator in both B and C phases. From these results together with the symmetry consideration, the gap function of UPt3 is determined as a E(1u) representation characterized by a combination of two line nodes at the tropics and point nodes at the poles.     

Selenacalix[4]dithienothiophene: Synthesis,Structure, and Complexation of a Cyclic Tetramer of Selenide‐Bridging Dithienothiophene

An efficient cyclization toward a cyclic tetramer of dithienothiophene (DTT) linked by divalent selenium atoms has been developed via palladium‐catalyzed coupling reaction of (nBu₃Sn)₂Se. X‐ray analysis revealed its highly symmetrical structure had an alternate arrangement of DTT units. There are several Se???π interactions forming a supramolecular network leading to large void channel space. The cyclic tetramer possesses moderate electron‐donating ability. Furthermore, the cyclic tetramer undergoes complexation with C₆₀ in a 1:2 ratio in the solid state to give a highly symmetrical three‐dimensional array of C₆₀.     

A Peanut‐Shaped Polyaromatic Capsule: Solvent‐Dependent Transformation and Electronic Properties of a Non‐Contacted Fullerene Dimer

Synthesis of molecular containers capable of incorporating multiple fullerenes remains challenging. Reported here is that room‐temperature mixing of metal ions with W‐shaped bispyridine ligands featuring polyaromatic panels results in the quantitative formation of a peanut‐shaped M₂L₄ capsule. The capsule reversibly converts into two molecules of an ML₂ double tube in response to changes in the solvent. Notably, the capsule allows the incorporation of two fullerene molecules into the connected two spherical cavities at room temperature. The close proximity yet non‐contact of the encapsulated C₆₀ molecules, with a separation of 6.4 Å, was revealed by X‐ray crystallographic analysis. The resultant, unusual fullerene dimer undergoes sequential reduction within the capsule to generate (C₆₀^.?)₂, C₆₀^.??C₆₀^2?, and (C₆₀^2?)₂ species. Furthermore, temperature‐controlled stepwise incorporation of two C₆₀ molecules into the capsule is demonstrated.