The many proofs of an identity on the norm of oblique projections

Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P ²=P, which is neither null nor the identity, it holds that ||P|| = ||I –P||. This useful equality, while not widely-known, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.     

光学薄膜在现代汽车工业中的应用

本文论述应用光学薄膜技术对汽车窗口玻璃和塑料的若干性能进行改进的技术发展水平，这些性能包括塑料表面的硬化，红外区的反射，紫外区的吸收，偏振化作用，双折射，憎水性以及光学角度选择性等，评论等离子体处理和物理蒸汽淀积薄膜工艺近来应用的一些例子。对于物理蒸汽淀积薄膜所提供的异常功能连同实际使用中它们的耐久性作了特殊的强调。     

Melting of heterogeneous vortex matter: The vortex ‘nanoliquid’

Disorder and porosity are parameters that strongly influence the physical behavior of materials, including their mechanical, electrical, magnetic and optical properties. Vortices in superconductors can provide important insight into the effects of disorder because their size is comparable to characteristic sizes of nanofabricated structures. Here we present experimental evidence for a novel form of vortex matter that consists of inter-connected nanodroplets of vortex liquid caged in the pores of a solid vortex structure, like a liquid permeated into a nanoporous solid skeleton. Our nanoporous skeleton is formed by vortices pinned by correlated disorder created by high-energy heavy ion irradiation. By sweeping the applied magnetic field, the number of vortices in the nanodroplets is varied continuously from a few to several hundred. Upon cooling, the caged nanodroplets freeze into ordered nanocrystals through either a first-order or a continuous transition, whereas at high temperatures a uniform liquid phase is formed upon delocalization-induced melting of the solid skeleton. This new vortex nanoliquid displays unique properties and symmetries that are distinct from both solid and liquid phases.     

Traces of functions with a dominating mixed derivative in ℝ<Superscript>3</Superscript>

We investigate traces of functions, belonging to a class of functions with dominating mixed smoothness in ℝ³, with respect to planes in oblique position. In comparison with the classical theory for isotropic spaces a few new phenomenona occur. We shall present two different approaches. One is based on the use of the Fourier transform and restricted to p = 2. The other one is applicable in the general case of Besov-Lizorkin-Triebel spaces and based on atomic decompositions.     

苯并菲盘状液晶的合成、亲氟效应及分子对称性对介晶性的影响

New symmetrical and asymmetrical triphenylene-containing discotic liquid crystals with two different peripheral alkyl chains, known as sym-TP(OC6H13)3(OR)3 and asym-TP(OC6H13)3(OR)3, were synthesized. Their thermotropic liquid crystal properties were investigated through polarizing optical microscopy (POM), differential scanning calorimetry (DSC) and X-ray diffraction (XRD) analyses. The asyrranetdcal discogens are 2,6,11-rialkoxy-3,7,10-trihexyloxytriphenylenes, with the alkyl chain carbon numbers varying from 3-10, 12, and 14, while the symmetrical compounds are 2,6,10-trialkyloxy-3,7,11-trihexyloxytriphenylene. Two fluoroalkoxy substituted triphenylene discogens, 2,6,10-td(4,4,4-trifluorobutoxy)-3,7,11-trihexyloxytriphenylene and its asymmetrical isomer 2,6,11-tri(4,4,4-trifluorobutoxy)-3,7,10-trihexyloxytdphenylene were prepared. These two compounds show higher melting and clearing points than their alkoxy analogs, which implies that fluorophilic effect exists in the formation and stabilization of discotic columnar mesophase. The triphenylene derivatives TP(OC6H13)3(OR)3 with two different peripheral chains, symmetrically or asymmetrically attached on triphenylene cores, have lower melting points and clearing points than those of the higher symmetrical compounds TP(OR)6 with the same total chain carbon numbers. The mixed-chain-triphenylenes with longer alkoxy chains (n=9,10,12,14) show columnarmesophase at room temperature.     

Functional Janus dendrimers containing carbazole with liquid crystalline,optical and electrochemical properties

ABSTRACT

Novel liquid crystalline Janus dendrimers that combine a mesogenic block and an electroactive block have been synthesised. The mesogenic block is based on two third-generation Percec-type dendrons bearing six or eight terminal dodecyloxy alkyl chains, whereas the electroactive blocks are formed by one or two carbazole units. The liquid crystal behaviour was investigated by polarised-light optical microscopy, differential scanning microscopy and X-ray diffraction. The Janus dendrimers with one electroactive unit exhibited cubic or columnar liquid crystal phases, whereas the Janus dendrimers with two electroactive units did not show liquid crystalline behaviour. The UV-vis absorption and emission properties of the Janus dendrimers were investigated. The spectra suggested the existence of π-π stacking and the formation of aggregates in the solid state. Electrodeposition of the carbazole-containing dendrimers afforded semi-globular particles in which the number of electropolymerizable units and the flexible or rigid character of the linker have a decisive influence in the particle size.     

Alkaloid production by callous tissue cultures of Cereus peruvianus (Cactaceae)

The morphologically undifferentiated cells of nonregenerant callous tissue of Cereus peruvianus cultured in the original medium and in medium supplemented with tyrosine were used as an alkaloid source. Comparison of alkaloid production by C. peruvianus plants and by callous tissues indicated that alkaloid levels were almost twice as high in callous tissues as in shoots of C. peruvianus plants. The ratio of alkaloid concentration between mature plant and morphologically und ifferentiated cells of callous tissue was 1∶1.7. A relationship between culture medium containing tyrosine and alkaloid production was also observed in the callous tissues of C. peruvianus. Since increased alkaloid production may be induced by additional factors such as tyrosine, increasing levels of tyrosine or other conditions of the culture medium may be considered factors for inducing higher alkaloid production by C. peruvianus callous tissues.     

Designing liquid-crystalline dendronised hexa-adducts of [60]fullerene via click chemistry

ABSTRACT

Liquid-crystalline [60]fullerodendrimers were constructed via click chemistry based on the reaction between hexa-adducts of [60]fullerene (C₆₀) bearing 12 azide groups and alkyne-terminated cyanobiphenyl dendrons of first- and second-generation. The structure of all the new compounds was confirmed by IR, UV, ¹H and ¹³C NMR spectroscopies and mass spectrometry. The mesomorphic properties were studied by polarised optical microscopy, differential scanning calorimetry and small-angle X-ray scattering. The hexa-adduct of C₆₀ functionalised with the first-generation dendrons gave rise to the formation of a smectic A phase and a rectangular columnar phase (c2mm symmetry) while the hexa-adduct of C₆₀ decorated with the second-generation dendrons displayed only a rectangular columnar phase (c2mm symmetry). Our results show that the hexa-adduct of C₆₀ is a unique synthetic platform for the design of fullerodendrimers and dendronised materials.     

Hexagonal columnar mesophases of polycatenar para-phenylene connected bis-oxadiazole-based liquid crystals

In this article polycatenar liquid crystals containing two 1,3,4-oxadiazoles interconnected by a para-substituted benzene ring as the central linking unit and three alkoxy chains at each terminal have been synthesised, and investigated by polarising microscopy, DSC and XRD scattering. Molecules with medium chain length form exclusively hexagonal columnar liquid crystalline phases.     

Enthalpically and Entropically Favorable Self-Assembly: Synthesis of C4h-Symmetric Tetraazatetrathia[8]circulenes by Regioselective Introduction of Pyridine Rings

Self-assembly of π-conjugated molecules in solution generally occurs owing to either an enthalpic or an entropic gain; however, designing π-conjugated systems that simultaneously exhibit enthalpically and entropically favorable self-assembly behavior is challenging. Herein, the self-assembly behavior of tetraazatetrathia[8]circulenes is disclosed, which is driven by both enthalpy and entropy. Single-crystal X-ray diffraction analysis demonstrated that molecules of these tetraazatetrathia[8]circulenes form face-to-face stacked dimers with a 1D columnar structure owing to the circularly arranged dipole moments. Importantly, concentration- and temperature-dependent ¹H NMR spectra revealed that the formation of self-assemblies of tetraazatetrathia[8]circulenes in chloroform and methanol is favored by both enthalpic and entropic factors. The unique association behavior is due to the presence of sp²-hybridized nitrogen atoms, which weakly coordinate to the hydrogen atoms of these solvents and reduce the π-electron density of the circulene cores.