A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]^–1T[(P)]<-12 max {T, P^–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]^–1T[(N)]<-12c² max {tT, N^–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.     

黑洞与奇点

黑洞可以说是引力最极端的体现,其视界内是个连光也逃不出去的时空区域。近来黑洞在天 文观测方面取得令人惊讶的发展,这其中包括：黑洞碰撞的引力波探测以及M87 星系的超大质量 黑洞的所谓第一张黑洞照片。但是在理论的层面上,黑洞物理尚有许多未解之谜。其中,信息遗失 的悖论是最有名的。但是,有另一个问题至少和信息的丢失一样{甚至更加{令人费解的,就是黑洞 内部的奇点性质。时空奇点是广义相对论本身无法描述的,在那里究竟发生什么事？黑洞内部的奇 点和宇宙大爆炸时的奇点有何不同？奇点是否会裸露在黑洞外面？所谓“宇宙监督猜想”的假设目 前有何进展？我们在这篇半科普的文章中简单的介绍这些课题,希望本文章对物理和数学的本科生 有所帮助。     

The effects of Cu doping on the magnetoresistive behavior of perovskites La0.7Ca0.3MnO3

The electronic and magnetic properties of Cu-doped perovskite La_0.7Ca_0.3Mn_1−xCu_xO₃ obtained by doping Cu on its Mn sites have been studied. The perovskite structure was found to remain intact up to the highest doping level of x=0.20. At low Cu concentration (x=0.05) the temperature-dependence of resistivity of the material exhibited up to two peaks corresponding to the magnetic transitions from the PM to the FM phase, and from the FM to the AFM phase. In general, the doping level was found to suppress the ferromagnetic ordering of the material, increase its resistivity, and produce large values of magnetoresistance near the resistivity peak. These results were explained as due to the formation of the antiferromagnetic phase.     

Determination of pharmaceuticals and related impurities by capillary electrophoresis

In the first part of this work, the use of capillary electrophoresis (CE) for the separation of two groups of pharmaceuticals, namely a metabolite of tamoxifen and a basic drug substance, DS1, was investigated. The effects of pH and types of modifiers, e.g. surfactant, bile salt, γ-cyclodextrin and hydroxypropyl-β-cyclodextrin on selectivity, separation and peak shape were studied. Besides achieving complete separation of the compounds, the CE system was capable of providing separation with significant improvements in overall peak shape of the compounds compared with HPLC. In the case of the basic drug substance DS1, validation of the CE system developed in terms of linearity, selectivity, sensitivity and reproducibility was satisfactorily performed. At the same time, a study of the sample solvent matrix effects on the separation of this group of compounds was examined. The system was successfully applied to the analysis of laboratory-synthesized samples. Good correlation was observed between CE and HPLC, although higher efficiency and faster speed of separation were obtained using the CE system developed. For the tamoxifen metabolite, special emphasis was placed on the use of CE for the separation of the pair of isomers. This was readily achieved through the introduction of γ-cyclodextrin in the electrolyte. Resolution of at least 1.5 was obtained for the isomers using the CE method.     

Cyclopolymerization. VI. Preparation and Properties of Crosslinked Polyamines by Cyclopolymerization

Crosslinked polymer and copolymers containing tertiary amino groups were prepared from a number of allylamino monomers by cyclopolymerization. The effects of the initiator residue, reaction conditions, monomer structure, and degree of crosslinking on specific properties such as ion-exchange capacity, pKa, and range of nitrogen basicities were investigated.     

Cyclopolymerization. VII. The 13C NMR Spectra of Cyclopolymers Obtained from N,N-Diallyamines

Pulsed Fourier transform ¹³C natural abundance nuclear magnetic resonance spectroscopy has been used to determine the structures of the polymers formed by the radical-induced cyclopolymerization of N-substituted-N, N-diallylamines and N-methyl-N, N-bls(2-alkylallyl)-amines. The polymers of N, N-diallylamines all contain cis-and trans-substituted pyrrolidine rings in the ratio 5:1. The polymers of N-methyl-N, N-bis(2-alkylallyl)amines give complex spectra due to the presence of both cis-and trans-pyrrolidine and -piperidine rings, but the difference in chemical shifts of the N-methyl signals from the different structural types allows the spectra to be analyzed.     

Amine‐Linked N‐Heterocyclic Carbenes: The Importance of an Pendant Free‐Amine Auxiliary in Assisting the Catalytic Reaction

We have successfully expanded the library of amino–NHCs with varying substituents on the amine group, leading to insight about the instability of NHCs arising from the intermolecular interaction of the dangling amine side‐arm. However, the pendant amine plays an important role with respect to the catalytic process, resuscitating the catalytic activity of unsaturated NHC’s through a synergistic effect invoked by the secondary amine. This proof of concept allows us to expand the spectrum of catalysis to C C, as well as C B bond formation.     

Biologically inspired 3Fe4S cluster as structural mimics of FeMoco M-cluster

A 3Fe4S cluster related to M-cluster of Mo-nitrogenase is reported. [K(THF)₅][Fe₃(μ-bdt)₂(μ-PPh₂)(CO)₅] ( 1 ) is synthesized from photo-assisted structural rearrangement of [K(THF)₂][(μ,κ²-bdt)Fe₂(μ-PPh₂)(CO)₅] under visible light irradiation. The molecular structure of 1 consists of a Fe₃ core bearing one dithiolate bridge and the second dithiolate group capping onto the metallic plane. The structural motif of 1 is analogous to that of Mo-participated 3Fe4S unit in M-cluster of Mo-nitrogenase, with the similar Fe-Fe and Fe-S bond distances. Upon protonation in 193 K, a Fe-hydride species ( 1H ) is generated and characterized to possess the bridging hydride group (δ = −12.57 ppm) by ¹H-NMR spectroscopy and DFT calculation. In CH₃CN solution, complex 1 exhibits a reversible reduction and oxidation process at $E_{1/2}^{\mathrm{red}}$ = −1.94 V and $E_{1/2}^{\mathrm{ox}}$ = −0.18 V, respectively, at 273 K. The reduction behavior of 1 in CH₂Cl₂ solution at 243 K displays a slight modification in the presence of trifluoroacetic acid, revealing a moderate anodic potential shift (~50 mV). The current amplitude of the reduction wave is linearly increased with the increasement of acid added, indicative of the catalytic event.